Where does the idea that faith must be a condition for baptism originate from?

After the apostolic writings, we find a continuation of instruction regarding informed baptism, rather than infant baptism.

Justin Martyr writing circa 150 CE is among the first to state faith/belief is first required and then baptism.

As many as are persuaded and believe that what we teach and say is true, and undertake to be able to live accordingly, are instructed to pray and to entreat God with fasting, for the remission of their sins that are past, we praying and fasting with them. Then they are brought by us where there is water, and are regenerated in the same manner in which we were ourselves regenerated. For, in the name of God, the Father and Lord of the universe, and of our Saviour Jesus Christ, and of the Holy Spirit, they then receive the washing with water. The First Apology Chapter LXI

Why is this not for babies?

And for this [rite] we have learned from the apostles this reason. Since at our birth we were born without our own knowledge or choice, by our parents coming together, and were brought up in bad habits and wicked training; in order that we may not remain the children of necessity and of ignorance, but may become the children of choice and knowledge, and may obtain in the water the remission of sins formerly committed, there is pronounced over him who chooses to be born again, and has repented of his sins, the name of God the Father and Lord of the universe; he who leads to the laver the person that is to be washed calling him by this name alone. -ibid-

Please note he claims the apostles taught this.

In addition, the Didache teaches baptism of adults and perhaps children, but not infants.

But before the baptism let the baptizer fast, and the baptized, and whoever else can; but you shall order the baptized to fast one or two days before. Didache Chapter 7

So, to answer the OP, the faith to believe and then be baptized originated with the apostles, compiled into the Bible, and carried into tradition within less than 80 years of their times.

Faith before baptism is hardly a new concept. Even before Jesus, baptism was performed on individuals that had repented of their sins and put their faith in God.

Consider these scriptures about baptism:

John the Baptist:

I baptize with water those who repent of their sins and turn to God. …
— Matthew 3:11

This messenger was John the Baptist. He was in the wilderness and preached that people should be baptized to show that they had repented of their sins and turned to God to be forgiven. … And when they confessed their sins, he baptized them in the Jordan River.
— Mark 1:4,5 (Also Luke 3:3)

… John the Baptist preached that all the people of Israel needed to repent of their sins and turn to God and be baptized.
— Acts 13:24

Jesus, after his resurrection:

Anyone who believes and is baptized will be saved. …
— Mark 16:16

Peter:

… Each of you must repent of your sins and turn to God, and be baptized in the name of Jesus Christ for the forgiveness of your sins. Then you will receive the gift of the Holy Spirit.
— Acts 2:38

… baptism, which now saves you, not by removing dirt from your body, but as a response to God from a clean conscience. It is effective because of the resurrection of Jesus Christ.
— 1 Peter 3:21

Philip:

But now the people believed Philip's message of Good News concerning the Kingdom of God and the name of Jesus Christ. As a result, many men and women were baptized.
— Acts 8:12

Paul:

The Lord said, “Go over to Straight Street, to the house of Judas. When you get there, ask for a man from Tarsus named Saul. He is praying to me right now. … Instantly something like scales fell from Saul’s eyes, and he regained his sight. Then he got up and was baptized.
— Acts 9:11,18

Crispus, the leader of the synagogue, and everyone in his household believed in the Lord. Many others in Corinth also heard Paul, became believers, and were baptized.
— Acts 18:8

… Paul … found several believers … they were baptized in the name of the Lord Jesus. Then when Paul laid his hands on them, the Holy Spirit came on them …
— Acts 19:1,5,6

For you were buried with Christ when you were baptized. And with him you were raised to new life because you trusted the mighty power of God, who raised Christ from the dead.
— Colossians 2:12

Clearly, before baptism, an individual must believe in the Gospel, repent of sin, and have faith to accept Christ's salvation.

Origin and category

Where does the idea that faith must be a condition for baptism originate from? It is easiest to simply study the history of the theology of baptism and by simply asking: which denomination started credo baptism, since prior to that infant baptism was universally practiced? The answer seems to be the first "Baptist" church (1609). See Wikipedia article Believer's baptism -- Protestant Reformation.

Categorization-wise, Lutheran, Anglicans, Methodists and Covenantal Calvinists practice infant baptism, along with Catholicism and Orthodox churches. The rest of the Protestants practice credo baptism. For the shades of theological meaning of infant baptism by different denominations, see Wikipedia article Infant Baptism -- Theology. The relevant parts of Christian theology are Sacraments, Regeneration, Faith, and Covenants.

For more understanding of your own view, one way is to see it from the lens of the other, more traditional, opposing view: infant baptism. Since believer baptism positions are very similar, I will provide the comparative Christianity introduction on infant baptism instead, and then refer you to 2 scholars in the Resources section.

Infant baptism, baptismal regeneration, faith, and public profession

"Clearly babies cannot believe in anything?" You're completely right. Denominations that practice infant baptism emphasizes community as a means of grace (sometimes informally said as a "sacrament"). For example, Reformed churches believe the preached Word is a sacrament, Catholicism teaches that the visible world-wide Catholic church IS a sacrament.

Your parents are God's agents to initiate you into Christianity, just like in the OT, 8-day old babies were circumcised so they become a "child of Abraham" and their parents need to instruct them in the faith (Deut 6:4-9). All denominations that practice infant baptism will insist that the parents of those babies need to promise to teach them the gospel. Faith comes later as the child becomes conscious of repentance and the need for Jesus as a result of those teaching. But the baptismal regeneration given when they were infants is the enabler (faith comes through grace) and also the objective sign and seal. Baptism is God's gift (Trinitarian life) and their parents's gift to them (initiation into the family of God).

To help see the role of faith in connection to infant baptism and to being born again (new life), I'll provide a short description of 3 main practices.

1. In the pre-Protestant "denominations" (Catholics, Eastern Orthodox, and other Orthodox churches) and possibly in Anglicanism too (I'm not so sure), baptism is seen as the sacramental instrument to convey the new life, so the infants are understood to receive the indwelling Trinitarian life already (no faith needed, baptismal regeneration is key). That's why catechism which preceded First Communion is important, so the child (being already regenerated) consciously understand their faith and exercise their faith helped by the grace already given at the infant baptism.

   In Catholicism (and possibly in the Orthodox churches), for the sacraments of Eucharist and Confession to become efficacious it is understood that faith is needed. Thus, faith needs to accompany sacramental practice, where one does the actions that help one to remain in the "state of grace":

   • preventively (by taking part in the Eucharist) and
   • reparatively (by confessing mortal sins before a priest, as one becomes aware of them, to receive absolution so they can be back in the "state of grace").

   If you are NOT in a state of grace you will not go to heaven.

   What if an adult is converting to Christianity and was never baptized before? In those denominations, the adult would already have faith before he was baptized, having understood the gospel and having received Jesus as his/her Lord and Savior, but the understanding of how baptism works for adults is STILL identical to that of infant baptism: faith is NOT the precondition to receiving the new life, but the sacrament itself (with God working behind the scene, of course, to regenerate the person). Usually in the same church service / mass, the adult will ALSO have confirmation and first communion, completing the initiation into the body of believers, which is also the adult convert's first acts of faith done in the new life given at baptism. Although in these denominations this is considered exceptions, for these converts, faith in Jesus CAN precede baptism as God cannot be bound by the sacraments!

2. In the Covenantal (non Baptist) Reformed churches (such as the one I grew up in), infant baptism is also considered as a sacrament BUT the baptism is only seen primary as the initiation into the covenant of grace and needs to be followed up with confirmation where the youth becomes a full professing Christian and is admitted to communion. I believe since Reformed theology emphasizes Election, Irresistible Grace, and Perseverance of the Saints, the exact moment where new life is given is not strictly tied to the infant baptism, to confirmation, or even to faith ! These denominations emphasize Monergism. Thus there can be Christians "in name only": they thought they are saved (having gone through the motions), but they are not. Thus in some Reformed circles, seeking the "assurance of salvation" can become a frightful preoccupation.

3. In denominations practicing a theology like Methodism (following John Wesley), there is this concept of prevenient grace, the grace which enables one to have faith in Jesus later when the child has personally repented and accepted Jesus as their savior (at which point where one receives justifying grace). In these denominations, the aspect of personal conversion and testifying about it is more prominent. If the child never comes to conscious faith then the "prevenient grace" given when he/she was an infant will not make him/her born again. Some prefer to delay baptism of their children until they can make the decision on their own; for them the understanding of baptism will be similar to your understanding.

Resources for further study

Baptism is an extremely complex and contentious topic. These 2 resources would be helpful to you:

• Gavin Ortlund (a credobaptist theologian and a Baptist pastor) who made the journey from paedobaptist to credobaptist wrote a short 2013 article Why I Changed My Mind about Baptism. He also have a recent (March 2022) video on Baptismal Regeneration: Responding to Common Arguments.

• Trent Horn (a Catholic apologist) had an extensive rebuttal of Gavin's video in Catholic Answers video shortly after (with transcript) Rebutting Gavin Ortlund on Baptismal Regeneration which I highly recommend as the resulting dialogue exposes the many aspects of baptism, shows differing interpretation of Scripture, quotes Patristic sources, and contain references to scholarly books on baptism for your further study.

• From the IVP Academic Spectrum Multiview book series: Baptism: Three Views.

• From the Zondervan Counterpoints book series: Understanding Four Views on Baptism.

Conclusion

Mixed denomination families become more common and can become a challenge. But as long as the family decides to emphasize what is common, such as settling on C.S. Lewis's Mere Christianity or on the evangelical emphasis in 1) being born again and 2) in the Lordship of Jesus, both parents can have different convictions of how exactly the new life is connected to baptism and to faith, while being gracious to one another to jointly select the denomination to raise their children in. It is not the denomination which saves, but Jesus. In all denominations, regardless of infant/believer baptism, Jesus will come through the various channels, given the desire for a Christian to belong to Him and His body. Parents can then ensure that their children have all these 3 elements by the time they reach adulthood: Trinitarian life (regeneration), Faith, and Baptism.

The simple answer is that, just as faith was required in the Old Testament for individuals to please God and to be in relationship with him, so in the New Testament, when baptism "took off afresh" for the nation of Israel with John the Baptist's baptisms, and with what Jesus Christ said, then with how all the apostles explained their agreement with faith needed for genuine conversion, water baptism then being a public declaration that such faith had been expressed.

A good list of 12 such N.T. texts has already been provided in another answer, so I will not repeat them here. However, the idea of credo baptism first originating in 1609 with a Baptist denomination is totally wrong, plus misleading. For a start, credo baptism is reported in the N.T! Thereafter, the Christian Church continued to practice it, with Tertullian condemning an instance of infant baptism in 197. Despite that, sprinkling of infants gained ground. Here is what this Catholic authority says on the matter:

"The baptism of adults is presupposed throughout the NT and the early Christian period... Infant baptism developed naturally out of the entirely different circumstances in which Christianity found itself when society had become Christian... But child baptism became the normal practice from about the 4th or 5th century onwards... even today, through the intermediary of their godparents infants are treated at their baptism as if they were adult: they [the godparents] renounce Satan, confess the faith, and state that they wish to be baptized." Encyclopedia of Theology, p69, article by Burkhard Neunheuser, Burns & Oates, 1961 [Emphasis mine]

This means that credo baptism was the single form of baptism until Catholicism approved this new practice later. It was based on the teaching that salvation was dependent on baptism, i.e. baptismal regeneration, (Sess. 7, Canons on the Sacrament of Baptism, 1-14, D 857-70) and that baptism could not be repeated. Groups that disagreed with this, saying salvation came first at the point of putting faith in Christ, were later declared to be heretics, and so the uprising of infant baptism caused them to be ousted from the Church.

There is written confirmation of this after 1348, but the fact that Catholicism declared credo baptism to be a heresy long before then shows that it was practiced by others. This means that trying to attribute credo baptism to either Anabaptists (twice-baptised) or to Baptists circa. 1609 is nonsense.

The new emperor, Charles IV, vigorously pursued (persecuted) all Christians who objected to such things as infant baptism. At that time, many people who had been baptised as babies or toddlers, later came to hear the gospel as proclaimed by Christian preachers outside of Catholicism, confessed Christ personally and were fully immersed in water. They were, effectively, the first Anabaptists, long before a particular group in Munster was named such, circa 1524. The Anabatist non-Christian excesses near the end of that century gave them a bad reputation, but their predecessors were virtuous Christians who would not accept Catholic dogma on such matters as infant baptism (and other issues.) They were all tarred with the same, one brush of 'Heretics'. All of them, from the 2nd century on, maintained that faith must be a condition of baptism. In this, they were simply sticking to the New Testament teaching, even though it cost many of them severe persecution, if not their lives. It predates Lutherism and Calvinism by centuries!