All inclusive resorts are a popular form of vacation for families, especially in the Caribbean, but you won't find them on every island. The list below indicates your best bets (such as the DR, Jamaica, Aruba, and St. Lucia.) Note that some other resorts do offer "inclusive plans" for meals and perhaps activities too; don't expect the utter freedom --of food, drink, activities-- of full-on all inclusive pricing, though.
Be aware that in some places, locals may be quite happy to to not have all inclusives, in the typical meaning of a large self-contained resort. Still, you may find a smaller property that offers an "inclusive plan."
(See Caribbean Island profiles for an introduction to top islands for families; use this map of the Caribbean Islands to orient.)
Things change, new resorts open-- but you're probably not going to find an all inclusive resort on this smaller, quieter island.
Antigua: Yes, you will find all inclusive resorts in Antigua; the luxury St. James Club and Villas has all inclusive pricing, welcomes families, and has a kids club. The Jolly Beach Resort is another family-friendly choice.
Aruba: this island has plenty of all inclusives; check some family-friendly choices.
Bahamas: you will find some all-inclusives in the Bahamas; a Viva Wyndham resort is one possibility. (Viva Wyndham is a value-priced and family-friendly resort brand; this property has a Viva Circus program, with trapeze lessons and more.) Also, a Breezes Bahamas welcomes kids 14 and up; Breezes is part of SuperClubs.
Barbados: not known for all inclusive resorts; Almond Village Resort is a popular large all inclusive resort, with kids club, family pool, and nursery; the Almond Casuarina Beach Resort is likewise AI and welcomes families.
Bermuda: Not much in the way of AI resorts.
Bonaire: Though not an all inclusive resort per se, Buddy Dive Resort welcomes families and has packages that include breakfast and diving.
British Virgin Islands: Bitter End Yacht Club & Resort likes kids and has packages that include meals, use of sailboats, kayaks, windsurfers, plus activities such as snorkeling trip.
Cayman Islands: visitors aren't likely to find an all inclusive resort option.
Curacao: All inclusive resort Breezes Curacao has kids programs.
Dominica: As Dominica is considered an "unspoiled" Caribbean island, its fans are probably glad there are no big all inclusive resorts. Adventurous families might enjoy Jungle Bay eco-resort where packages are pretty inclusive: i.e., meals and activities such as hikes, ocean kayaking, snorkeling.
Jamaica: AI pricing is dominant, and both the Sandals/Beaches and the Superclubs/Breezes chains have their base in Jamaica. See Jamaica All Inclusive Resorts. And unique to Jamaica: at Franklyn D. Resort, your stay includes your own vacation nanny.
St. Kitts and Nevis: These tiny islands (lush, upscale) don't feature AI resorts but do have two prominent family-friendly choices, (Four Seasons Nevis, and St. Kitts Mariott.)
Turks & Caicos
Families flock to the all inclusiveBeaches Resort here. (Beaches is the family brand of the Sandals chain).
US Virgin Islands: Sugar Bay Resort and Spa is all inclusive, a rarity in the USVI; Bolongo Bay Beach Resort has an all inclusive plan. Also, the Divi Carina Bay All Inclusive Beach Resort & Casino, on St. Croix, has become an all inclusive resort.
Remember, change happens! New resorts get built, existing ones add new pricing options...