"If you're only doing good because you're afraid you'll be punished by God if you don't, you're not a good person". This is a common response to the moral argument. This, of course, is not an accurate representation of the argument. The moral argument is not that without God, the Christian making the argument would stop doing good things, but that without God, moral absolutes do not exist, and therefore regardless of what I would and wouldn't do if I stopped believing in God, and regardless of what you do and don't do as an atheist, all moral claims are merely a matter of opinion.
If I stopped believing in God today, I would not become an axe murderer tomorrow. I don't find it especially unsurprising that most atheists are also not inclined to murder people. After all, I do believe God has written the law on our hearts (Romans 2:11-16). We all have a natural moral compass, even if it is deteriorated.
But supposing, for a moment, one of us did become an axe murderer. Who's to say that's morally wrong? It's one thing to say "I don't like axe murderers", but what sets your personal preference above another person's? Atheists are quite happy to acknowledge this fact when it comes to issues they agree with. For example, they make memes that say "if gays can't get married because of your religion, you can't have a cookie because I'm on a diet." Well, if I can't chop people up with an axe because you don't like it, you can't have a cookie because I am on a diet. It's the same principle. With no God, there are no moral absolutes, and therefore no atheist can really claim to be a good person, nor can they really condemn evil. The absolute best they can say is "I try to conform to the standard I have personally decided upon".
But still the atheist insists they are "good without God". They're not "good" because God tells them to be, they're "good" because they do what they call good (this is important. Even Hitler believed he was a good person) without being told.
So, let's just ignore the fact that atheism cannot account for moral absolutes. We will pretend that the moral views we have in common are possible to sustain in an atheistic worldview. Here's the problem: If one can be considered "good" by doing good without being told, one can also be considered evil by doing evil without being told. And yet, with the exception of a few extremely arrogant individuals, most people live by the mantra "nobody's perfect". To err is human, and boy do we err.
This includes Christians, by the way. It's not like you convert and boom, you're Jesus. No, every Christian sins. If you could line up every human who has ever existed in order of goodness, there would probably be a Christian between every pair of unbelievers. So yes, every Christian sins.
So back to the question: If we can be called good for doing good without being told, can we not also be called bad by doing bad without being told? Let's go deeper. Let's go to the layer none of us really want to go. Which persuasion supersedes the other? Does bad invalidate good, or does good invalidate bad?
I like to use the very sad example of Temar Boggs. Temar Boggs was a teen hero. When a child was dragged into a car, he chased the kidnappers on his bike until they stopped and released the child. He even showed humility, stating that he believed it was what anyone would have done. This is very good.
But Boggs' heroism soon came to an end when he instead participated in a robbery, which, if memory serves, was armed. He was caught, he was taken to court, and of course he was convicted. His previous act of heroism was actually brought up by his lawyers in an attempt to gain sympathy. This failed. I ask it, did the courts make a bad call here?
I have no doubt that some people would say yes. The depravity of man knows no bounds, and so there are probably those who believe you should be able to get away with murder in some circumstances. But the reality is that evil always eclipses good. A little leaven leavens the whole lump. A little pollution makes water undrinkable. Even a single rotten apple soon takes over the whole bunch. In a good courtroom, no amount of good will erase the crime, and God's courtroom is perfect. Because of your sin, God calls even your righteous deeds "filthy rags" (Isaiah 64:6).
So, when you meet Him, will He call you a good person and let you into Heaven, or a bad person and send you to Hell? Only one person in all of history could ever claim the former. Everyone else, me, you, we all sin, and so God's only option is to declare us "guilty". Your good deeds are irrelevant.
But here's the thing. Our trial has already passed, and our sentence has already been served. Jesus Christ, the Son of God, came to earth, lived a perfect life and died on the cross. He didn't die for His sins, but for the sins of the world. That means God has already punished the sins you commit. So, you can be free. You can repent of your sin, you can confess Jesus as Lord, and if you believe in your heart that God raised Him from the dead, you can be declared perfect, and you can inherit eternal life. Or you can reject Him, receiving God's wrath for your sin yourself. The latter is a complete waste. The former is available to you for as long as you breathe.