This all is based on the citizenship of your country your passport is issued from, even if youl live in the US now.
For US Citizens, most caribbean countries don't require a visa or a passport to travel to them. Why? because most Americans don't have passports and they don't want to miss out on getting you to spend your tourist dollars there.
They don't require Visas (pre issued stamps/certificates to enter their country) to US citizens, but they might if were from some 'non favored' country, but this is all different per island you go to.
The new Passport requirement to travel to these countries that is mandated that everyone have passports for all travel by the end of this year is a US requirement. You can prob legally still go with a birth certificate, but you will have trouble re-entereing the US. Because of this, airlines aren't going to let you depart in the first place unless you've got the proper travel documentation to start with. I've seen them deny boarding because of this before to someone. (They just don't want to have to mess with the problems with a person who isn't properly documents. Now days also they would consider it a security concern).