Buildings that allow pets appeal to ALL buyers and as a result, increase the buyer pool and property values. Buildings that do not allow pets drastically decrease their buyer pools because anyone who has a pet will not even look in these buildings. Encourage your board to allow pets, install a dog run (if possible), and write up ground rules for pet owners. The board may want to have a size limit of fewer than 50 lbs and encourage owners and renters to pay a pet deposit.
I've been selling apartments for 30 years, and my answer is definitely YES. since so few buildings allow pets I think it would help. My experience is that even in no pet buildings plenty of people have pets, so you may as well. Once suggestion is to charge a "pet fee" to cover any messes