The Caribbean's economy undergoes a complete restructuring that would actually do it some good in the long term. While initially there'd be a massive economic crash (probably a population one too as less slaves get imported) and a relegation to a backwater, eventually new investments would see the region flourish once more.
For instead of just the insanely profitable sugar for most islands, you now get plantations that focus on other cash crops such as tobacco, quinone, citrus, Indonesian spices and probably cotton. Additionally, ranching, logging, shipbuilding, artisanal work and other forms of small scale light industry.
Big winner here in the short to mid term would be Spanish Hispaniola, as despite being the neglected stepson of the Spanish colonies it had a diversified economy from which sugar was a small part of its GNP. While other colonies have their economies crash Santo Domingo wouldn't even blink.