Should the university be popular? It wasn't in the middle ages where town and gown battles could even result in bloodshed. In more recent times universities have been outposts of radical and liberal dissent; they also in some places have been the home of ultranationalism and Fascism.
In America, universities, like churches have been free of most forms of taxation. Should we regard recently approved taxes on university endowments as a sign of popular hostility (or at least of Republican hostility)? In my university experiences, as a student and for almost 40 years as a professor at a respected state university, I found that in the non-scientific areas there was indeed a liberal bias, but that was a long time ago and it still exists. I do not know firsthand how Princeton faculty are selected; I do know from my experience in this process in my department of history that scholarship was more than sometimes evaluated on the basis of ideology and politics rather than on scholarly grounds, something I always found distressing. The public knew and knows nothing of this but it knows the result without knowing the process: Most humanities and social science faculty are on the left. Of course some should be on the right for balance and fairness, but I don't know that this would make a real difference in popular perceptions.
I prefer that the university remain unpopular, even if finances thereby take a hit. But I close by affirming that for every tax dollar wrenched from a university one hundred dollars should be wrenched from the churches who do far less good and far more harm than any university, no matter how undistinguished, could ever do.