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.