Newborn and cat

So I am half way through this pregnancy and now I am kinda starting to stress about my cat and a newborn. I know my dogs will be fine because one sleeps with my daughter and the other in his kennel (he was abused and that's where he feels safe). So we now have a cat, I have never had a cat. I have heard they will climb in the crib and climb on the baby. So how do you prevent this? We don't shut bedroom doors so what did you ladies do? Sorry if this is stupid to some and no getting rid of my cat is not the answer. All my animals are my babies also! Just curious what others do to handle the cat and baby situation? Thanks