First, what about Christ? Did he sin? Obviously not! So we have to concede that there is at least one exception to this notion that "there is none who does good." But if there is even one exception, then we have to concede that verses like this simply cannot be read literalistically.
But there is more. Because if "there is none who does good" - literally - then not even the apostles could be said to have done good! So even if we're willing to create the admittedly special exception for the Son of God - which would be a reasonable exception, of course - we're either going to have to say that the apostles never did good, or else there are still other exceptions.
But there is more. Because even if we grant the apostles as an exception (because they were, after all, apostles), we still have other problems. Because, for example, St. Peter says that "the eyes of the Lord are upon the just, and his ears unto their prayers" (1Pet. 3:12). How can this be if there is literally none who does good? But perhaps we must create an additional exception for Christians.
But even this is insufficient. For St. Paul writes of the Jewish people, "I bear them witness that they have zeal for God" (Rom. 10:2). Their zeal is hindered by the fact that they are ignorant of things that they ought to know, but they have zeal for God nevertheless. Well, is zeal for God a bad thing? Clearly St. Paul does not think so! So again we see that we must create an exception for the possibility that even non-Christians may do good.
But someone might say that the Jews are a special case. But even this is not sufficient. Because as we know, the Lord Jesus Christ said of the Roman centurion, "Amen I say to you, I have not found such great faith in Israel" (Mt. 8:10). Now clearly the Lord believed that this man's faith was a good thing, and yet he was not a Christian, and he was not a Jew. He was a Gentile.
And what about the man who helps the little old lady across the street? Does he not do a good deed? What about the fellow who risks his own life to save the life of someone else? Is this not a good deed? Of course it is!
So now we have come full circle: we have seen that Christians and Jews and Gentiles are all capable of doing good, and in fact actually do good things, and so we are forced to conclude that Romans 3:12 and 3:10 cannot reasonably be interpreted in a literal way. Hence those Protestants - particularly those Reformed - who like to say that no non-Christian does good are flatly wrong.
But at least some Reformed folks will want to make a distinction here, perhaps, and it seems that it is a reasonable one. They will say that the good things that we do are insufficient for us to merit salvation. And in a certain sense, this seems to be reasonable. "For if Abraham was justified by works, he has reason to boast, but not before God" (Rom. 4:2). None of us deserves by our own merits to go to heaven. Rather, we receive the gift of salvation by the grace of God, through the death and resurrection of our Lord Jesus Christ.
Of this Justification the causes are these: the final cause indeed is the glory of God and of Jesus Christ, and life everlasting; while the efficient cause is a merciful God who washes and sanctifies gratuitously, signing, and anointing with the holy Spirit of promise, who is the pledge of our inheritance; but the meritorious cause is His most beloved only-begotten, our Lord Jesus Christ, who, when we were enemies, for the exceeding charity wherewith he loved us, merited Justification for us by His most holy Passion on the wood of the cross, and made satisfaction for us unto God the Father; the instrumental cause is the sacrament of baptism, which is the sacrament of faith, without which (faith) no man was ever justified; lastly, the alone formal cause is the justice of God, not that whereby He Himself is just, but that whereby He maketh us just, that, to wit, with which we being endowed by Him, are renewed in the spirit of our mind, and we are not only reputed, but are truly called, and are, just, receiving justice within us, each one according to his own measure, which the Holy Ghost distributes to every one as He wills, and according to each one's proper disposition and co-operation. For, although no one can be just, but he to whom the merits of the Passion of our Lord Jesus Christ are communicated, yet is this done in the said justification of the impious, when by the merit of that same most holy Passion, the charity of God is poured forth, by the Holy Spirit, in the hearts of those that are justified, and is inherent therein: whence, man, through Jesus Christ, in whom he is ingrafted, receives, in the said justification, together with the remission of sins, all these (gifts) infused at once, faith, hope, and charity. For faith, unless hope and charity be added thereto, neither unites man perfectly with Christ, nor makes him a living member of His body. For which reason it is most truly said, that Faith without works is dead and profitless; and, In Christ Jesus neither circumcision, availeth anything, nor uncircumcision, but faith which worketh by charity. This faith, Catechumens beg of the Church-agreeably to a tradition of the apostles-previously to the sacrament of Baptism; when they beg for the faith which bestows life everlasting, which, without hope and charity, faith cannot bestow: whence also do they immediately hear that word of Christ; If thou wilt enter into life, keep the commandments. Wherefore, when receiving true and Christian justice, they are bidden, immediately on being born again, to preserve it pure and spotless, as the first robe given them through Jesus Christ in lieu of that which Adam, by his disobedience, lost for himself and for us, that so they may bear it before the judgment-seat of our Lord Jesus Christ, and may have life everlasting (Council of Trent, Sixth Session, Decree on Justification, Chapter VII).So: of course our salvation comes from Christ, and no one would pretend that you can (for example) live in adultery and expect to gain salvation because you saved a baby from a fiery death. But the fact that one sins by (for example) adultery does not take away from the objective good of (for example) saving a baby's life.
Yes, people do good. No, they cannot save themselves apart from Christ.
I can imagine at least one or two practical applications of this. For one thing, we need not assume (as some certainly seem to do) that non-Christians' motives are evil. Anti-Catholics should not assume that Catholics' motives are evil. Catholics should not assume that the motives of anti-Catholics are evil. In the absence of actual evidence to the contrary, the law of charity ought to govern how we treat others at all times: that is, we ought to grant them the benefit of the doubt. This is all the more essential when it comes to matters of the heart. We do not even know our own hearts. How then can we judge the hearts of others? We can't. This is not to say that we must blind ourselves to evil. But we really ought to be more cautious in assigning blame. A sin is a deliberate action that someone takes. If it seems that someone lies about us, we need to be able to prove that in fact at the time he made the false statement, he was both aware of the truth and specifically intended to say something untrue instead. If we do not know this - and we rarely (if ever) do know such a thing (certainly we can hardly know that at all on the Internet) - then we have an obligation to be charitable.