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.