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.
It's hard to say whether America has become more conservative over the last 50 years or not, because depending on the President, the country can shift from being more liberal to more conservative. 50 years ago would be 1963, with the beginning of the Lyndon Johnson adminstration. Johnson was a major force behind domestic issues as his "Great Society" programs took action on civil rights, poverty, welfare, health, and topics of that sort. These would all be considered more aligned with the liberal ideology. Even the Presidents following Johnson were moderate Republicans, in Nixon and Ford. Nixon even proposed a liberal idea by means of the EPA.
It really wasn't until Reagan that American moved more towards conservatism. Reagan was the ultimate conservate, seen through his strong faith, and his priorities of increased military spending and tax cuts. Though after Reagan we've had two Republican Presidents, no one has matched his level of conservatism.
As for the present day, I would say America would not be considered more conservative than it was 50 years ago. For instance, as time has passed, more and more citizens have come in support of gay marriage and marijuana, things that would have been unheard of in the 1960s. Those two issues are things that liberals favor more than conservative. In addition, the thought of having an African American president would have been unheard of five decades past. Environmental issues and talks about global warming and divestment been more spotlighted in the past decade, and a lower percentage of Americans than before are associated with a religion. With that said, while it's debatable whether America has become more liberal, I definitely would not say it has become more conservative.