This is, of course, a matter of opinion. There are ways in which the US has become more conservative over the past 50 years and ways in which it has become more liberal. Which trend has been more dominant is a matter of opinion.
In some ways, the US is much more liberal today than it was in late 1963. This is particularly true when it comes to social issues. In 1963, the idea of gay rights, let alone of gay marriage, would have been utterly absurd. In 1963, the idea that an African American might be president would have been laughable. Similarly, the idea that many people think a woman (Hilary Clinton) might be the next president would have struck most people as impossible. Today, all of these things are seen as normal and acceptable (though gay marriage has not gained as much approval as racial and sexual equality).
In other ways, though, the US has become more conservative. In the 1960s and 1970s, there was not the same kind of push to limit government spending that there is today. In those years, people were much less antagonistic towards the federal government than they are now. The highest marginal tax rates were much higher than anyone would accept today.
Thus, there have been changes towards greater conservatism in the last 50 years, but also changes towards greater liberalism. I would say that we are much more liberal in social terms but somewhat more conservative in economic terms.