Barbados. It is very safe compared to a lot of other Caribbean islands. It also has some of the best health care in the Caribbean, if anything ever went wrong. You could not pay me to take my child to Jamaica.
For comparison -
Barbados:
http://travel.state.gov/travel/cis_pa_tw/cis/cis_1022.html
Jamaica:
http://travel.state.gov/travel/cis_pa_tw/cis/cis_1147.html