What's the biblical basis for non-believer's ability to do good?

Three things must be kept in mind:

1. Everything good is from God, as such good things done by anyone (believer or non-believer) comes only from God.

James 1:17: Every good gift and every perfect gift is from above, and cometh down from the Father of lights, with whom is no variableness, neither shadow of turning.

2. Spirit of God is striving with men (whether good or evil) calling them to repentance. Sometimes bad people do respond.

Genesis 6:3 And the LORD said, My spirit shall not always strive with man, for that he also is flesh: yet his days shall be an hundred and twenty years.

A good example would be the story of Balaam. Even after knowing that God disapproves the course of his action, he went to curse Israel. However even in the midst of the wicked course, he respond to the spirit of God and desires to die the death of the righteous:

Numbers 23:10: Who can count the dust of Jacob, and the number of the fourth part of Israel? Let me die the death of the righteous, and let my last end be like his!

3. Good in the sight of God is very different from good in the sight of men. In God's sight anything done with even a little selfishness or a little side gain, does not amount to good.

Donating a good amount of money for a specific cause might be a good action in men's sight, but God judges motives, and only He can tell if such an action is really good.

Matthew 6:5: And when thou prayest, thou shalt not be as the hypocrites are: for they love to pray standing in the synagogues and in the corners of the streets, that they may be seen of men. Verily I say unto you, They have their reward.

So in conclusion:

1. Bad people can do good things but even that comes because they hearken to the voice of God in their hearts because God is constantly calling men to repentance and Good

2. Not every good action is really good.

This story comes to mind. Remember the story of the good Samaritan? The moral of this story answers your question. No matter what religous building you belong to in faith, even if you don't consider yourself a religous person at all, you have the freewill to choose how you will handle life's situations. Even as a non believer you can still love your neighbor and do good. Sadly sometimes its the opposite as well. When people such as priests or others in places of authority that do believe, but choose to not do good.

Luke 10:25-37King James Version (KJV)

25 And, behold, a certain lawyer stood up, and tempted him, saying, Master, what shall I do to inherit eternal life?

26 He said unto him, What is written in the law? how readest thou?

27 And he answering said, Thou shalt love the Lord thy God with all thy heart, and with all thy soul, and with all thy strength, and with all thy mind; and thy neighbour as thyself.

28 And he said unto him, Thou hast answered right: this do, and thou shalt live.

29 But he, willing to justify himself, said unto Jesus, And who is my neighbour?

30 And Jesus answering said, A certain man went down from Jerusalem to Jericho, and fell among thieves, which stripped him of his raiment, and wounded him, and departed, leaving him half dead.

31 And by chance there came down a certain priest that way: and when he saw him, he passed by on the other side.

32 And likewise a Levite, when he was at the place, came and looked on him, and passed by on the other side.

33 But a certain Samaritan, as he journeyed, came where he was: and when he saw him, he had compassion on him,

34 And went to him, and bound up his wounds, pouring in oil and wine, and set him on his own beast, and brought him to an inn, and took care of him.

35 And on the morrow when he departed, he took out two pence, and gave them to the host, and said unto him, Take care of him; and whatsoever thou spendest more, when I come again, I will repay thee.

36 Which now of these three, thinkest thou, was neighbour unto him that fell among the thieves?

37 And he said, He that shewed mercy on him. Then said Jesus unto him, Go, and do thou likewise.

One way to think about man's corruption is taintedness.

If a glass full of absolutely pure water has added a small quantity of extremely fine ricin powder (a virulent poison), the water is no longer fit for drinking because it is tainted, and the glass of water as a whole is not good. However, the water in the glass is still water and as water, is good per se. If one were to drink the tainted water, it would indeed quench thirst—but within a few hours, serious symptoms would occur, and within a few days, death. The tainted water possesses goodness, but not the kind that matters to the question of drinking it. There is no part of the water, as is, that can be safely drunk. It is wholly tainted.

Similarly, as all have sinned and fall short of the glory of God, everyone is wholly tainted by sin. But being wholly tainted is not the same as being wholly taint. There is still some goodness in all of us. Given that we are made in the image of God and we are able to make choices, then so long as the question is not our ultimate position before a holy God but our temporal actions in the world, we have the capacity for some goodness:

Matthew 7:11 NASB

If you then, being evil, know how to give good gifts to your children, how much more will your Father who is in heaven give what is good to those who ask Him!

When it comes to our ultimate position before God, our own acts of goodness are still tainted, and there is no wholly altruistic act possible—every sip of water from our glass has some amount of deadly poison in it, completely ruining it for drinking. Nonetheless, there is still real goodness in us. Certainly, caring for one's children is good, even if it is done for somewhat selfish reasons.

Jesus's statement that "no one is good but God alone" was to highlight the wrongness of the rich young ruler's beliefs about his own goodness, which had prompted Jesus's "why do you call me good?" question. It wasn't an absolute theological position about our glasses being full of only ricin and no water. Jesus said what he did to expose the man's beliefs about goodness as a whole.

People are sadly mistaken when they claim that "I'm basically a good person" if they mean the kind of goodness that would positively affect any absolute moral judgments on their persons, such as if God were considering whether they were guilty of sin. If the question under consideration is whether they are "good to drink"—and being tainted, they are not—then in the grand scheme of things, even the most minuscule amount of taint equals a complete failure to meet God's standards.

Just as Jesus was pointing out to the rich young ruler, having a form of goodness common to the world is not sufficient to inherit eternal life, because the taint prevents the glass from being suitable for its ultimate intended purpose—intimate freely loving union with God for all eternity.

A pastor I once knew said that he thinks of people as "tainted ladders." A ladder is good for something: you can climb up to the roof of your house with one, for example. But ladders aren't suitable for reaching up to God. They are too short—the chasm between us and God is infinite, because we are tainted.

To pre-answer possible objections and give some possible clarity about related topics: I see our initial act of repentance as not a good work, precisely because it is the cessation of an evil work. People are born with the innate characteristic such that when they come face-to-face with a holy God (and a holy God's testimony about Himself), their natural response is to fall on their faces in abject humility and healthy fear, keenly knowing their unworthiness. It is only people's evil and continuously repeated acts of the will to harden their hearts and resist the truth/light that enables them to live outside of right relationship with God at all.

God draws all people. His power and invisible attributes are clearly shown by what has been made. He convicts all people of sin via the Holy Spirit. And He either holds people accountable to what they know or He offers prevenient grace by softening their hearts enough to cause them to see past their own ignorance and blindness.

God doing all these things brings people to a place where they can see themselves more truly, but doesn't install faith or change their hearts to the point where the heart decision to believe God has been made for them. Each person is the one who is responsible for whether or not he/she comes to faith. And this does not make the moment of acquiring saving faith a work that saves—each person "gives up the battle" and believes God's testimony about himself, and this is not a work—God then does all the (good) work of saving, in response to the condition He Himself set for such saving. God is bound to save those who come to faith because He promised to do so. God has the right and ability to set conditions on salvation, and did, and then always acts in accord with His own character and promises.

Surrender to God is not an act but a ceasing to act—a giving up of oneself to the former opponent, for Him to do with as He pleases, to have a right relationship with Him and with the truth (and with sin). This surrender is also done in recognition of the justice of doing so, and with the hope that the stronger combatant's mercy will yield a better result than continued, losing, unjust, military opposition. It's "WHOA! I'm dealing with the God of the universe here. Resistance truly is futile. I'm on the wrong side, and I've committed great evil, and I'm done for!" It isn't the intellectual "I think that I'm going to switch sides in this battle as a calculated strategy to win." There is no winning out over God or freedom from constraint nor is there self-ownership. There is only rebellion or submission to Him; the decision whether to be a slave to sin or a slave to righteousness.

Our reformed brethren who are Calvinists would likely take exception to my statements. To them, Total Depravity, the T of TULIP, really means Total Inability—they believe that the unregenerate can do no good whatsoever, not even respond to God, not even recognize their own depravity when convicted by the Holy Spirit, not even accept their condition and God's reality that God has shown them. They believe that the act of coming to believe in God, if it is the result of a human's own choice from his own nondeterministic and self-arising initiation, would be a self-saving good work rather than faith which is not a work, therefore (they say), people people cannot choose to come to faith, only God can choose to bring people to faith.

I honestly don't find that in the Bible nor in my personal experience of reality.

The biblical basis for man ‘doing’ or ‘wanting’ to do or be ‘good’ is actually Genesis chapter 3. By ‘eating’ from the tree of the knowledge of god and evil, man was effectively deciding to be his own source for or of righteousness. The serpent said ... ‘you will be like gods’, and, that was not a lie - as attested in verse 22 of chapter 3. Man ‘instinctively’ wants to responsible for his own righteousness, that is, ‘be good’. Man is instinctively self righteous. So the basis is in the fall.

Full outline found in ‘A God of Grace and Mercy - Stoned to Death for Picking up Sticks’ by HdH (Amazon).

Before answering this question, I need to point out a few problems with the verses you supplied in support of your thesis.

Genesis 6:5 had to do with a particular point in time and applied to a particular population on earth at the time; it was never meant to refer to all peoples at all times.

I believe Isaiah 64:6 is referring to the Jewish people. I don’t think Isaiah is including Gentiles when he uses “we” in the chapter. He is just being deeply ashamed of his nation’s state before God.

The passage that does support your thesis is Romans 3:11-12; however, it could be extended as Romans 3:9-18, where Paul asserts that “Jews and Greeks … are all under sin.”

In the eyes of man, people can indeed do good things. This is undeniable. But, according to Romans, in the eyes of God, no one is good enough to merit righteousness through any deeds they do. The unrighteous will do good deeds to one another, yet without ever being able to earn a reward from God for them (Matthew 5:46-47).

Unregenerate people can do good things, but they’ll never be righteous inside through their deeds. One (the works) is of the outward; the other (righteousness) is of the inward. God looks on the inward for true worth; and He can’t find it among the unregenerate because only what is done to honor Christ by faith in Him is righteous. The unregenerate honor themselves, not Christ.

On this basis, I see no biblical basis for an unregenerate person to do what is righteous in God's eyes.

The original post states "On the contrary, scripture seems to show that humanity is incapable of doing anything good" Romans 3:11-12 is given.

Consider:
The Greek word used for "good" in Rom 3 is 5544. chréstotés. That greek word is only used in Romans when talking about God's goodness.
The Greek word used for "good" in Rom 2:7 is 18. agathos. That greek word is talking about man's good deeds.
Finally, Galatians 5:22 has both Greek words, and distinguishes them as Kindness vs Goodness.

I believe scripture does not say 'men cannot do good deeds' (see Rom 2:7 which states the opposite). I believe it says man cannot make himself good.
Romans 3:12 is a quote from Psalms 14:1.
There, "does good" are two Hebrew words which might be translated "make good".
The Hebrew for "does" is the same word used in Genesis 1:7 to describe God "making" (the expanse).

I think this sums up to prove that man can do good, but he can't make himself good in the way God is good.

The problem seems to stem from a radical divergance between the commonly understood meaning of 'good' and the sense it is used in certain parts of the Bible - particularly the verses you cite as well as:

"Why do you call me good?" Jesus answered. "No one is good--except God alone. - Luke 18:19 NIV

Rather than (as per Webster's) being "of somewhat high but not excellent quality", Jesus is using the term here to describe the intrinsic perfection of God's nature to always do what is exactly right - the only way people can be considered 'good' in this sense is if they become "partakers of the divine nature" (cf. 2 Peter 1:4).

That said, perhaps the clearest biblical passages that addresses whether 'non-believers' (actually they still believe in something, even if it is that their - God-given if unrecognised as such - conscience is a reliable guide) can do 'good' (in the common sense) are:

So God created mankind in his own image,

in the image of God he created them;

male and female he created them. - Genesis 1:27 NIV

and

6 God “will repay each person according to what they have done.” 7 To those who by persistence in doing good seek glory, honor and immortality, he will give eternal life. 8 But for those who are self-seeking and who reject the truth and follow evil, there will be wrath and anger. 9 There will be trouble and distress for every human being who does evil: first for the Jew, then for the Gentile; 10 but glory, honor and peace for everyone who does good: first for the Jew, then for the Gentile. 11 For God does not show favoritism.

12 All who sin apart from the law will also perish apart from the law, and all who sin under the law will be judged by the law. 13 For it is not those who hear the law who are righteous in God’s sight, but it is those who obey the law who will be declared righteous. 14 (Indeed, when Gentiles, who do not have the law, do by nature things required by the law, they are a law for themselves, even though they do not have the law. 15 They show that the requirements of the law are written on their hearts, their consciences also bearing witness, and their thoughts sometimes accusing them and at other times even defending them.) 16 This will take place on the day when God judges people’s secrets through Jesus Christ, as my gospel declares. - Romans 2:6-16 NIV (emphasis added)

In summary, though mankind in general has fallen short of God's standard of perfection, because we have been made in God's image and he has written his law on our hearts, when we do actually act accordingly, those particular works can be considered 'good' (at least in the common sense of the word) in and of themselves - the fall marred the image of God, and our consciences have been darkened, but the image is not completely destroyed, and our consciences do still work to a degree. The Bible of course makes the big call that there is no-one that does actually follow their conscience consistently - rather that our pre-redemption nature is inclined predominantly to evil instead.

The Bible recognises the good that King Cyrus of Persia did for the Jews, historically. Not just in allowing them to return to Judah, but in funding the considerable cost of rebuilding the Jerusalem temple. For example, Isaiah 41:2 refers to Cyrus as the righteous man from the east, and Isaiah 44:28-45:4 waxes lyrical:

That saith of Cyrus, He is my shepherd, and shall perform all my pleasure: even saying to Jerusalem, Thou shalt be built; and to the temple, Thy foundation shall be laid. Thus saith the LORD to his anointed, to Cyrus, whose right hand I have holden, to subdue nations before him; and I will loose the loins of kings, to open before him the two leaved gates; and the gates shall not be shut; I will go before thee, and make the crooked places straight: I will break in pieces the gates of brass, and cut in sunder the bars of iron: And I will give thee the treasures of darkness, and hidden riches of secret places, that thou mayest know that I, the LORD, which call thee by thy name, am the God of Israel. For Jacob my servant's sake, and Israel mine elect, I have even called thee by thy name: I have surnamed thee, though thou hast not known me.

Of course, non-believers do not feel the need for biblical recognition that they can perform good deeds with good intent.