What is the biblical basis for the Athanasian Creed's statement that believing in the Trinity is necessary for salvation?

TLDR; It's easy to prove the answer is "The case cannot be made directly from scripture."

Now, it's discouraged here to answer a question other than the one asked, but I'm going to do a bit of that here to demonstrate why the answer must be "No."

1. If such a basis exists, it's not found directly in scripture

   Even the most superficial research will show that the word Trinity is not found in scripture. And while the concept of Trinity is arguably in scripture, it's not plainly so. It took the church until the late 4th century to reach anything resembling an agreement on the doctrine in its present form.

   As such, even the strongest Trinitarians don't cite direct scriptural evidence that the Trinity even exists. It took literally centuries to reason out the doctrine from the clues found in scripture.

   Therefore, any doctrines building upon the Trinity must also come from this line of reasoning, using scriptural concepts to build upon them logically. This means that any basis for the idea that believing in the Trinity is necessary for salvation cannot come directly from scripture, since the idea of the Trinity itself does not come directly from scripture.

2. One cannot write about that which one does not understand

   Since the concept of the Trinity was formed centuries after the books of the Bible were authored, it thus follows that the Biblical authors didn't have the concept of the Trinity in mind when writing their texts. Perhaps they knew there was an interesting, complex relationship between the Son and Father, between the Son and Holy Spirit, etc., but the concept of the Trinity had not yet been formulated, and therefore could not be written about.

   If the authors of the Bible did not know of the Trinity, how could they write about it being a requirement for salvation? If they can't write about this requirement, it cannot be found in the pages of the Bible.

1. You must believe that God took on human form in the person of Jesus.

   1 John 4:2-3

   2 This is how you can recognize the Spirit of God: Every spirit that acknowledges that Jesus Christ has come in the flesh is from God, 3 but every spirit that does not acknowledge Jesus is not from God. This is the spirit of the antichrist, which you have heard is coming and even now is already in the world.

2. You must acknowledge that Jesus is the Son of God, hence there is a father that is God.

   1 John 4:13-15

   13 This is how we know that we live in him and he in us: He has given us of his Spirit. 14 And we have seen and testify that the Father has sent his Son to be the Savior of the world. 15 If anyone acknowledges that Jesus is the Son of God, God lives in them and they in God.

   (Also, recall the Lord's Prayer, "Our father, who art in heaven...". It is instructive that Jesus taught his disciples to pray to the Father, and that this prayer, above all others, has been a permanent and prominent fixture among all Christians since the beginning of our faith.)

3. You must receive the Holy Spirit, the urgency of which was recognized by the Apostles.

   Acts 19:

   1 While Apollos was at Corinth, Paul took the road through the interior and arrived at Ephesus. There he found some disciples 2 and asked them, “Did you receive the Holy Spirit whena you believed?”

   They answered, “No, we have not even heard that there is a Holy Spirit.”

   3 So Paul asked, “Then what baptism did you receive?”

   “John’s baptism,” they replied.

   4 Paul said, “John’s baptism was a baptism of repentance. He told the people to believe in the one coming after him, that is, in Jesus.” 5 On hearing this, they were baptized in the name of the Lord Jesus. 6 When Paul placed his hands on them, the Holy Spirit came on them, and they spoke in tongues and prophesied.

4. Proof that you know the Son is that you know the Father, hence can distinguish between Father and Son and know both of them.

   John 14:

   6 Jesus answered, “I am the way and the truth and the life. No one comes to the Father except through me. 7 If you really know me, you will know my Father as well. From now on, you do know him and have seen him.”

   There are many statements in the Bible from which one can infer the Trinity, but the above statements are among the more imperative declarations.

Addendum:

To address an important omission, I add Jesus' statements that both demonstrate that the Father is God (He is greater than all), the Son is God (Jesus claims to grant eternal life), and that the two are one ("I and the Father are one)". This passage further declares that the people he is addressing do not believe this, hence are not part of his flock - they aren't saved.

John 10:25-30

25 Jesus answered, “I did tell you, but you do not believe. The works I do in my Father’s name testify about me, 26 but you do not believe because you are not my sheep. 27 My sheep listen to my voice; I know them, and they follow me. 28 I give them eternal life, and they shall never perish; no one will snatch them out of my hand. 29 My Father, who has given them to me, is greater than all[; no one can snatch them out of my Father’s hand. 30 I and the Father are one.”

Acts 16:31 states

They replied, “Believe in the Lord Jesus and you will be saved, you and your household.”

And Acts 10:43 says,

About him [Jesus] all the prophets testify, that everyone who believes in him receives forgiveness of sins through his name.

Similarly, John 3:16 says

For this is the way God loved the world: He gave his one and only Son, so that everyone who believes in him will not perish but have eternal life.

And John 3:36 states,

The one who believes in the Son has eternal life. The one who rejects the Son will not see life, but God’s wrath remains on him.

Likewise, Mark 16:15-16 states,

He said to them, “Go into all the world and preach the gospel to every creature. The one who believes and is baptized will be saved, but the one who does not believe will be condemned.

Because of these verses, belief in Jesus Christ is widely regarded as a requirement for salvation, and the biblical basis arises from the interpretation of what, exactly it means to "believe" in Christ. This statement in Acts 16:31 and similar statements throughout the Bible make what you believe about who Christ said he was extremely important. In Knowledge of the Holy, A.W. Tozer frames it this way, (pg 5)

Among the sins to which the human heart is prone, hardly any other is more hateful to God than idolatry, for idolatry is at bottom a libel on His character. The idolatrous heart assumes that God is other than He is - in itself a monstrous sin - and substitutes for the true God one made after its own likeness. Always this God will conform to the image of the one who created it...

Not all who called themselves Christians through the centuries were Trinitarians, but as the presence of God in the fiery pillar glowed above the camp of Israel throughout the wilderness journey, saying to all the world, “These are My people,” so belief in the Trinity has since the days of the apostles shone above the Church of the Firstborn as she journeyed down the years. Purity and power have followed this faith. Under this banner have gone forth apostles, fathers, martyrs, mystics, hymnists, reformers, revivalists, and the seal of divine approval has rested on their lives and their labors. However they may have differed on minor matters, the doctrine of the Trinity bound them together.

This, however, is the real Christian faith, that we honor one single God in three Persons and three Persons in one single Godhead..

this tenet of the ancient creed has been held by the Eastern and Western branches of the Church and by all but a tiny minority of Christians.

And for this reason, belief in the Trinity is not just important, but is required for salvation. If Jesus really claimed to be God, then If one does not believe that Jesus was a member of the trinity, a part of the triune Godhead, then one does not actually believe in Jesus or his claims. Instead they believe in an idol of their own making, for it is clear in scripture and it has been clear throughout history to so many theologians that Jesus claimed to be God. As St. Thomas Aquinas said in Nature and Grace,

Explicit belief in the Trinity has therefore been necessary for salvation from the very beginning. ...it is impossible to believe explicitly in the mystery of the incarnation of Christ without faith in the Trinity. For the mystery of the incarnation of Christ includes that the Son of God took flesh, that he made the world new through the grace of the Holy Spirit, and that he was conceived by the Holy Ghost.

Introduction: A full explanation for the biblical basis of the doctrine of the Trinity can be found here: https://irr.org/biblical-basis-of-doctrine-of-trinity

Because the OP asks for the biblical basis for claiming that belief in the Trinity is necessary for salvation, I will not attempt to prove the biblical basis for the Trinity but simply refer the reader to the article above by the Institute for Religious Research (May 2011). It is a lengthy document that cites about 1,000 references from over 300 different chapters of the Bible, including references from all 27 books of the New Testament.

Parts I and II of the study begin by establishing the fact that there is only one God, the Lord. Part III shows from the Bible that the Father of our Lord Jesus Christ is God; Part IV that the Son, Jesus Christ, is God; Part V that the Holy Spirit is God. Part VI explains how the Father, Son, and Holy Spirit are each distinct from the other two and Part VII summarises how the Bible teaches the Trinity. For a quick review of the importance of the Trinity, read Part VIII: https://bib.irr.org/biblical-basis-of-doctrine-of-trinity-part-viii-what-difference-does-trinity-make

The objective of what follows is to establish the biblical basis for the necessity of accepting the Trinity doctrine. If we reject the deity of Jesus, the Son, and the deity of the Holy Spirit in their roles within God’s eternal plan of salvation, then how can we expect to be saved?

With regard to the role of Jesus Christ in salvation, here is a short extract from Part IV, section C 2, which shows that the title of Saviour, which belongs in the ultimate sense only to God, is ascribed to Jesus as the source of salvation. Remember there is only one true God who is called Saviour – yet this title is held by Jesus.

Today in the town of David a Saviour has been born to you; he is Christ the Lord (Luke 2:12)

With regard to the Samaritans who became believers in Jesus, “now we have heard for ourselves, and we know that this man really is the Saviour of the world” (John 4:42).

Not only is Jesus our Saviour, Jesus is also Lord:

and you will receive a rich welcome into the eternal kingdom of our Lord and Saviour Jesus Christ (2 Peter 1:11).

But grow in the grace and knowledge of our Lord and Saviour Jesus Christ. To him be glory both now and for ever! Amen” (2 Peter 3:18).

Yet 1 Timothy 4:10 declares that the first Christians put their hope in the living God, who is the Saviour of all men. How can it be then, that Jesus is the source of salvation?

he became the source of eternal salvation for all who obey him (Hebrews 5:9).

Part IV, section 5 is also worthy of your consideration with regard to the role Jesus plays in salvation. It can be summed up in John 5:21:

For just as the Father raises the dead and gives them life, even so the Son gives life to whom he is pleased to give it. Moreover, the Father judges no-one, but has entrusted all judgment to the Son, that all may honour the Son just as they honour the Father. He who does not honour the Son does not honour the Father, who sent him. Source: https://bib.irr.org/biblical-basis-of-doctrine-of-trinity-part-iv-son-jesus-christ-god

With regard to the role of the Holy Spirit in salvation, Romans 8:1-27 spells out His role in no uncertain terms. It is the Spirit of life that sets us free from the law of sin and death. It is the power of the indwelling Holy Spirit that transforms the believer. The believer is controlled by the Spirit:

And if the Spirit of him who raised Jesus from the dead is living in you, he who raised Christ from the dead will also give life to your mortal bodies through his Spirit, who lives in you (Romans 8:11).

Acts 5:3-4 clearly establishes the biblical truth that to lie to the Holy Spirit is to lie to God. 2 Corinthians 3:17-18 confirms that the Holy Spirit is the Lord:

Now the Lord is the Spirit, and where the Spirit of the Lord is, there is freedom. And we, who with unveiled faces all reflect the Lord’s glory, are being transformed into his likeness with ever-increasing glory, which comes from the Lord, who is the Spirit.

The Holy Spirit is Eternal (Hebrews 4:19), was involved in Creation (Genesis 1:2; Psalm 104:30; Job 33:4); was responsible for the Incarnation (Matthew 1:18, 20; Luke 1:35); Intercedes for Christians with the Father (Romans 8:26).

The Holy Spirit is sent by Christ to convict of sin, speaks not on his own but on behalf of Christ, and always glorifies Christ. Christians must be filled with the Spirit (Ephesians 5:18). His role in our salvation is described in John 16:7-14. This is no mere source of power or energy – this is God at work in the process of salvation. Source: https://bib.irr.org/biblical-basis-of-doctrine-of-trinity-part-v-holy-spirit-god

Conclusion: The OP seems to have no problem in accepting the incarnation, that God dwelt with us in human form as Jesus, yet fails to understand the role of the Holy Spirit in salvation. One way of looking at it is to realise that just as Jesus was God with us back then, so too the Holy Spirit is God with us now. Father, Son and Holy Spirit are inextricably linked in the work of salvation. Without acknowledging who Jesus Christ REALLY is and without understanding who the Holy Spirit REALLY is, believers are in spiritual darkness.

Sometimes it takes a long time before the scales are removed and the believer finally understands the eternal relationship between Father, Son and Holy Spirit. But for those who cling to the notion that before Jesus came to earth he was either created or born as a mighty spirit creature, and who think the Holy Spirit is just some sort of impersonal source of energy from God, then they are spiritually blind.

Only by understanding the roles all three play within the Godhead can Christians give worship, praise and glory to the one true God, to worship Him in spirit and in truth. That is why the Athanasian Creed declares, without apology or excuse, that “we worship one God in Trinity, and Trinity in Unity; Neither confounding the Persons; nor dividing the Essence. . . . He therefore that will be saved, let him thus think of the Trinity.”

The full Athanasian Creed can be found here: https://www.crcna.org/welcome/beliefs/creeds/athanasian-creed

Caveat: The Athanasian Creed demands belief in all of its tenets for salvation. While in agreement with the tenets, I do not believe that all of them are mandatory for salvation.

Explanation needed: When I first came to saving faith in Christ Jesus I did not believe in the Trinity doctrine. In fact, I was anti-trinitarian. But that did not prevent God from responding to my heart-felt repentance and my plea to be forgiven for my sins. The Holy Spirit confirmed to me that I had been adopted into God's family. The spiritual heart transplant happens before the heavy-duty theological stuff can get into your head. God never expects his children to run before they can walk, spiritually speaking. It took time for the reality of who Christ Jesus and the Holy Spirit REALLY are to sink in. But God was patient with me (as He is with all who sincerely repent and step out in faith), and the message got through later.

Having said that, I now realise that to deny the eternal relationship between Father, Son and Holy Spirit results in a wrong understanding of the plan of salvation. It's not enough to say you believe in the Incarnation yet exclude the role of the Holy Spirit. There is only one true God who is composed of Father, Son and Holy Spirit, and that is why "we worship one God in Trinity, and Trinity in Unity; Neither confounding the Persons; nor dividing the Essence."

Jesus himself equates salvation with knowing God:

Now this is eternal life: that they know you, the only true God, and Jesus Christ, whom you have sent. - John 17:3 NIV

Elsewhere, the scriptures negate the idea that this could be a mere intellectual knowledge about God:

You believe that there is one God. Good! Even the demons believe that—and shudder. - James 2:19 NIV (cf. in context)

Instead, examining the preceding context (cf. John 14, 15 & 16) we must understand this knowing of God in terms of a relationship that includes intimate familiarity with His nature, which is revealed in the persons of Father, Son and Holy Spirit and consequently includes "believing in the Trinity":

⁷ If you really know me, you will know my Father as well. From now on, you do know him and have seen him.”

⁸ Philip said, “Lord, show us the Father and that will be enough for us.”

⁹ Jesus answered: “Don’t you know me, Philip, even after I have been among you such a long time? Anyone who has seen me has seen the Father. How can you say, ‘Show us the Father’? - John 14:7-9 NIV

¹⁵ “If you love me, keep my commands. ¹⁶ And I will ask the Father, and he will give you another advocate to help you and be with you forever— ¹⁷ the Spirit of truth. The world cannot accept him, because it neither sees him nor knows him. But you know him, for he lives with you and will be in you. ¹⁸ I will not leave you as orphans; I will come to you. ¹⁹ Before long, the world will not see me anymore, but you will see me. Because I live, you also will live. ²⁰ On that day you will realize that I am in my Father, and you are in me, and I am in you. - John 14:15-20 NIV

I no longer call you servants, because a servant does not know his master’s business. Instead, I have called you friends, for everything that I learned from my Father I have made known to you. John 15:15 NIV

When the Advocate comes, whom I will send to you from the Father—the Spirit of truth who goes out from the Father—he will testify about me. - John 15:26 NIV

¹³ But when he, the Spirit of truth, comes, he will guide you into all the truth. He will not speak on his own; he will speak only what he hears, and he will tell you what is yet to come. ¹⁴ He will glorify me because it is from me that he will receive what he will make known to you. ¹⁵ All that belongs to the Father is mine. That is why I said the Spirit will receive from me what he will make known to you. - John 16:13-15 NIV

²⁹ Then Jesus’ disciples said, “Now you are speaking clearly and without figures of speech. ³⁰ Now we can see that you know all things and that you do not even need to have anyone ask you questions. This makes us believe that you came from God.” - John 16:29-30 NIV

In summary, in much of his extended discourse following the last supper, Jesus teaches his disciples regarding the persons of the Trinity with the expressed intent that they may know God; and in the midst of his "High-priestly prayer" in chapter 17, he equates knowing this God (ie the Trinity) with eternal life.