Definitely want to check with each country you plan to visit. After proving all vaccinations are up to date and a recent (within 6 mos) health certificate from a vet, I'd imagine most would let you keep the dog as long as it doesn't get off the boat, but if you plan on taking it ashore, it'll probably have to go thru quarantine period (15-30 days), don't think any country will let you just pull in, jump off the boat, and run the beaches with your dog. But, just a WAG based on helping other folks ship dogs state to state and internationally when I was a vet tech..
If you are a broke sailor now, cruising the Caribbean with a dog will make you broker. It can be quite expensive with required visits by vets, possibility of quarantine in some places. I don't think they will accept your assurance that the boat will not leave the boat. People certainly do it, but it does add a level of complexity.
It's often a pain but it varies dramatically by country. Lots of countries in the Caribbean now want you to apply in advance for an import permit for the dog instead of just showing up with generic papers. Often once of the documents required is a health certificate endorsed by a government official, not just signed by the vet. Sometimes this health certificate with be an export certificate from the last country.
Getting the timing right for the various requirements can be a challenge since they often want it done within 2 weeks of your arrival and they'll take several days to send you the permit after receiving everything.
Expect to require more different vaccines and to do them more frequently as well as a rabies titer test (which isn't cheap or quick).
Just figuring out the rules in various places is a challenge (like, Dutch St Maarten just changed their rules to require an import permit but can I check in on the French side without one?)