We charge $20 per night per dog, and collect a fully refundable $250 security deposit per dog. Our maximum is 2 dogs. Guests are asked to complete and sign a copy of our Dog Agreement (contract). We have several restrictions - no puppies under 1 year, dogs must be spayed or neutered (less likely to mark, to be aggressive, to roam, often are more calm, etc.), dogs must be well behaved on leash, and owners must agree to keep dogs on leash whenever they are outdoors. We have lots of wild critters here - bear, deer, turkeys, fox, eagles, coyotes, etc., and the leash rule is intended to protect wildlife as well as the dogs and our neighbors. It is amazing how few problems we've had. Vacuums are available to guests at both of our homes and we tell guests if they clean up after the dog/s and there are no damages, we'll refund 100% of the deposit. I don't know how many visiting dogs we've had over the years, but it's a lot. The owners are thrilled to be able to bring them along on vacation, and we have found that dog-owning guests are often better than those without pets. In 5 years we have only withheld money once from the deposit. We have dogs ourselves and allowing them at the rentals is really a win-win.
Sorry this is so long, but I want to add a comment about the new restrictive policy Airbnb has implemented. If you check the 'pets accepted' box, their default is 5 animals (!!! right??) and you're required to enter a per-stay fee. The host has no flexibility to charge per night, instead of per stay.
My workaround is to message guests immediately after booking with a copy of our Dog Agreement. The Agreement contains information about our fees and deposits. I bill through the Resolution Center and, after the stay, I promptly refund the deposits. Not ideal, but there doesn't seem to be any other way to do it.