We have a cat, and the only issue we had was that the cat assumed the crib was his and he would go in there to sleep when DD wasn't in there. We got him out of that habit tho before DD went into the crib at 3 months.
We bought a cat net and didn't end up using it! We put up his crib before he was born and just told him off when he went in it. By the time DS was born he knew it wasn't a place for kitty's!