Among the group of mathematicians that I know best, category theorists, a common attitude is as follows.  (G)CH is neither true nor false.  It's something that a model of (any given collection of) axioms for set theory might or might not satisfy --- and that's that.  So being "pro" or "anti" doesn't make sense.  If there were a God-given model of set theory then we could ask whether (G)CH was true in it, but there isn't.

(I'm probably projecting here, but even if this isn't a majority opinion among category theorists, I'm pretty sure it's a common one.)

I'm not aware of any work in category theory that depends on (say) a topos satisfying or violating (G)CH.  My ignorance doesn't mean much, though.