Americans are very positive about the Constitution in general, but not about some of the things that are actually in it. For example, here's a question from a Gallup Poll taken in January 1941: "Do you think newspapers should be allowed to take sides in their editorials during election campaigns?" 59% said yes, 33% said no, and 7% were undecided.
But more educated people are more likely to favor civil liberties, or at least say that do, and that's the case here--about 80% of college graduates said it should be allowed. So given the increase in educational levels, we might reasonably expect support to be much higher today. There are no recent results, but the question was asked in 1985. At that time, only 45% said that newspapers should be allowed to take sides, 48% said they shouldn't, and 7% didn't know.