While it is true that I am a bit bias on this subject, I completely understand landlords that do not want to allow pets in their rental property. Pets can be very messy, especially if their owners don't stay on top of making sure everything is cleaned and well maintained. Property owners with carpet also have an extra concern with pet stains and claws that can cause the rug to need replacing, which is an extremely expensive process if done correctly. Animals can tend to smell if they are not taken care of, and this is true whether they are in a cage or if they are given free rein in the apartment. While a responsible pet owner will not allow this to happen, there are far too many people who have pets that are a bit less than fully responsible.
At the end of the day, whether or not you allow a pet is up to you as the owner of the property. Some landlords will only allow certain kinds of pets, like fish or lizards. Others will only allow pets that stay strictly indoors, such as anything in a cage or indoor cats. Even still, other people will only allow animals under a certain weight and size in their apartments. The fact is that you have the decision about whether or not you will let your residents get an animal. You may even request a waiting period so that you can see what kind of resident they are before making that decision. As long as you are upfront about the policy you have, everything should work out fine.