The Amsterdam Manor Beach in Oranjestad advertises itself as pet friendly. It's all inclusive for food so maybe a bit on the expensive side but looks nice. I would call to see about costs and restrictions for pets.
I can't imagine trying to bring a pet with me. Do you have to or is it a personal preference? I don't have any dogs or cats right now but I would probably be looking for a nice pet hotel while I am away if I did have.