Two particularly intriguing points raised by Bernstein and the comments to his post: (1) David suggests that declining anti-Catholic feeling is partly due to the convergence of Catholic and non-Catholic beliefs and practices:
Post-Vatican II, the Catholic Church is less foreign, both in prayer (in that mass is now in English), sociologically (because Catholics no longer differ that much from other Americans in where they send their kids to school and how many children they have), ...
The economic literature on religion suggests that losing distinctive characteristics is a bad thing from the point of view of maintaining a church as an organization. If the price of reduced prejudice is indeed assimilation in terms of changed practices, that could be a very high price to pay.
(2) as one of the comments to David's post noted, " if nothing else this thread is going to demonstrate that anti-Catholic sentiment is alive and well." Many of the comments to Bernstein's post are interesting, but far more than I expected do reveal some degree of anti-Catholic prejudice.
One could argue that the dimished cultural stigma of being Catholic can help people consider it with more purity, being free from the cultural baggage that it's associated with.