USD are widely accepted in the Caribbean for taxis, shops etc. Just make sure the locals are quoting you in USD and not local dollars. Pounds are not easily accepted, unless you can change them on the ship or a bank.
We take US dollars in 20's,10's and 5's larger values are often trouble to use and the islands will often give you change in local currency. Sterling is a good backup and can be change on cruise boats for dollar.
US$ accepted everywhere.