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Ramil Карибский бассейн refers to the archipelago. One can say я живу на Карибских островах. In the official media they would say В странах Карибского бассейна. But usually when one wants to refer to the place simply a country's name will be used:
Carribean countries are: Aruba, Barbados, Bonaire, the Cayman Islands, Saint Croix, The Bahamas or Antigua, islands of Cuba, Dominica, Hispaniola, Jamaica, Montserrat, Puerto Rico, Saba, Saint Kitts, Saint Lucia, Grenada, Saint Vincent, Guadeloupe, and Trinidad & Tobago. One of these names will be used I think.