Even if you take a slip in a marina, as long as you certify on your inbound clearance form that the animal will remain aboard at all times, you shouldn't have any problems.
As Doghouse says, the islands with no rabies are very strict. In Bequia, a cruiser told me that he took his dog ashore for a "potty break", and a customs
official saw him. The official told him to take the dog back to the boat, and never bring him back to shore, or he would shoot the dog on sight.
When we moved to Nevis
, it was harder to get our cat into the country than it was for the two of us and all our furniture. It took a solid three months to jump through all the hoops of shots, tests and paperwork, and then we had to keep the cat quarantined in our house with weekly visits from the government
vet to make sure she was healthy.
The islands that have rabies present do not restrict you from landing your pet. But be very careful. On some islands, locals will put out poison for dogs
to eat. They think they're protecting their livestock (goats and sheep) by doing that.