As I said in my last post, very few surveys have asked about elites or elitism. But back in 1996, a Gallup/CNN/USA Today poll asked whether the description "The Party of the Elite class in this country" applied more to the Democrats or the Republicans. The results: 65% Republican, 19% Democrats, 4% both, 4% neither, 8% don't know. Opinions differed by political views, but even among extreme conservatives, more people said Republicans than Democrats (32% to 30%).
But political commentators seem to have a very different impression--many of them are convinced that the Democrats are really the party of the elites. For example, in the New York Times Book Review a few weeks ago, Christopher Caldwell (a conservative) says that 19 out of the 20 richest Zip codes "gave the bulk of their money to the Democrats in the last election", while Jonathan Alter (a liberal) says it's only 8 out of 10.
Of course, 1996 was a while ago, and maybe popular opinion changed since then. But the gap between public opinion and what might be called elite opinion on the subject is still impressive.