The good thing for you is that you have options. You may put your foot down and say no pets what so ever. Other landlords say that they will allow pets as long as they are under a certain size. Some landlords will only allow fish or small lizards but nothing with fur or that will get out of a cage. I saw a listing recently that would allow any animal as long as it fit in a bag that you could carry. While I am sure that some people will take advantage of that to allow animals through that the poster did not intend, it is certainly an interesting way of putting the pet policy. At the end of the day, you do have the final decision but how should you prepare to make it?
One of the most important things to think about is the type of flooring that you have in your apartment. Many landlords will not allow animals with fur if they have carpet in their property. While some people may think this is ridiculous, they will still have to respect your decision. Hard wood flooring is a bit more forgiving with some things than carpet, but this can also be said of human residents as well. This is especially true if the humans have a baby.
At the end of the day, you get to make the decision about having a pet in your apartment. Think to yourself though, if you are a pet owner, whether or not you would want to be somewhere without your cat or dog or snake or whatever it is you own. I think that is the best way to make your decision.