keeping cats out of crib?

I'm 8 1/2 months pregnant with my first baby. My hubby and I are planning on having the baby's crib near our bed, simply because we don't have that much room to put it elsewhere. We also have 2 cats, who love to sleep pretty much anywhere soft. I'm worried that once the baby comes, our bigger cat might try to snuggle up to the baby and suffocate it (mom being paranoid?), since she loves to snuggle up to us and sleep ON us. I also don't want cat hair, dander and litter where the baby will be sleeping. So we're thinking of building some sort of "lid" for the crib, which won't restrict the baby at all, but keep the cats out.

Am I worrying for nothing? We've started teaching the cats not to sleep in the bed with us, hoping it'll carry over to the crib, but I'm not sure that's best way to go about it. One cat gets it, and won't get on the bed when we're there, but the other cat (the larger one I'm a bit more worried about to start with) just doesn't get it... she's just too social!

I would just the close the bedroom door and the cats would have to
go sleep elsewhere. I would start training them by closing your bedroom
door and make this a cat free zone. Just my opinion.

I agree close the door and keep the cats out. My sister just had a baby and has two cats (one very social) and she has found him in the crib. So the cats aren't allowed in the nursey and in their bedroom when the baby is in there.

My mother bought us a crib tent (you can google it), but we never used it because our cats gave the baby wide berth, lol, but who knows, your kitties might want to snuggle with the baby. Congratulations on your upcoming parenthood! Buckle up 'cause it's a wild (but great) ride!

I guess I should have mentioned this, but forgot... we don't have a bedroom door. We live on the 3rd floor of a house, and the only 2 rooms we have up there with doors are the bathroom & laundry room... I guess we could confine them to the laundry room, but I can't imagine they'd be very happy!

I have seen many many people train cats just using a baby gate
that they are not allowed in or out of room. Seems weird as we
know they can jump such a small height.