Why is hydrogen sulphide being generated on Caribbean beaches?
It would seem from a quick perusal of the literature that this is caused by sulphate-reducing micro-organisms. To quote from the eponymous Wikipedia article:


  Sulfate occurs widely in seawater, sediment, or water rich in decaying organic material. Sulfate-reducing microorganisms are common in anaerobic environments where they aid in the degradation of organic materials.


which is consistent with the situation described.

In biochemical terms, these anaerobic bacteria are using sulphate rather than oxygen as the final electron acceptor (oxidizing agent) in the elctron transport chain which generates ATP through the chemiosmotic mechanism. The reaction for the reduction of the sulphate to hydrogen sulphide (dissimilatory sulphate reduction), according to the Wikipedia article already quoted, is:



A simple way of thinking about this is that in both aerobic and anaerobic respiration the oxidizing agent itself becomes reduced. Whereas oxygen is reduced to H2O, sulphate is reduced to H2S.