As others have mentioned, gay men are not necessarily more cultured. There are many more straight men at the opera or at museums, etc. at any given time than there are gay men, IME. (But I'm making an assumption there on what is meant by "cultured".) Culture is not a sexually-determined trait.
As Cunning Stunt mentions, it's a generalization (like gays make better hair dressers/florists/designers, etc.). But many generalizations/stereotypes tend to have a grain of truth to them. Exploring how these stereotypes develop can be worthwhile, though.
Honestly, I've given this a little thought and come up with and shot down several theories as to where this may have originated. The only one I can think of is that in large urban areas, like New York or Washington or SF, gay men were often employed to act as "walkers" - men who would accompany women to social functions (theater, opera, etc.) when their husbands could not or did not want to. This preserved the woman's honor and allowed her to attend a public function. Generally, the orientation of the walker would have been known to the woman's group and, for quite some time, it was considered scandalous for a married woman to attend public functions alone.