Does modern American liberalism stem from Puritan beliefs?

To a limited extent, yes. Puritans believed in the power of the individual conscience against tyranny and repression. Modern-day liberals believe much the same thing, except that their idea of moral conscience is secular, rather than religious.
Puritans and liberals can also be said to share a common belief in an ideal society. Puritans wanted to establish a kingdom of the godly on earth, a society purged of all godlessness and sin. Whereas liberals also tend to aim high in terms of the kind of society they wish to see, a society which has a similarly utopian flavor. Liberals, no less than their Puritan forebears, tend to see politics in terms of what's right and wrong, and there's a strong moral component to the kind of policies they endorse.