Actually, Cuba, Haiti, DR, Puerto Rico, and Trinidad all have fully-fledged cities. But otherwise I agree with amccarth that you should determine which islands you're most comfortable traveling to with your infant.

WIth my family, we've traveled to Aruba, St. Maarten, Antigua, and Virgin Gorda with chidlren under the age of two. We needed medical care on one occasion when one of the kids got a heat rash, which could have been taken care of on any of those islands.