I've allowed tenants to rent with dogs sometimes. Having been a dog trainer I have different worries about that than most landlords. I'm more worried about noise and/or fearful aggressive dogs than destruction. Dog owners often don't know their dog has separation anxiety and barks continuously while the owner is gone. These dogs are also the ones most likely to damage a unit. Dog owners often won't admit even to themselves that their dog is a fear biter. Both of these issues make it more likely that you'll have turnover of other tenants. Many places limit the size of the dog allowed, but I think it's the temperament that is more important.
Here's what I've done:
1. Allow dogs that are AKC certified Canine Good Citizens. That means that a dog trainer certified by the AKC has deemed the dog to have not shown any temperament issues during the test, that the dog knows some basic commands,and is friendly with strangers and other dogs. This also shows you that the owner is responsible. Anyone who goes to the trouble of getting the CGC is knowledgeable and involved in training their dog.
2. extra pet deposit and extra pet rent and a detailed pet addendum to the lease
3 . require renters insurance in the amount of XXX and add me as an additional insured
4. meet the dog before hand
5. make sure your insurance company doesn't have any breed restrictions and if it does exclude those - and know that many pit bulls are called lab mixes