The Atlantic hurricane season is officially from 1 June to 30 November. Peak time is June - Sept.
If you are going on a cruise check out this site http://www.cruisecritic.com/ - scroll down a bit to see 'Read Members Reviews' - then you can read honest varied opinions on your chosen ship. Then click on the word 'Boards' at the top and you can see lots of info on your cruise line and on your ports of call. You do have to be a member to see some items - it is free and it is worth it as this site is invaluable mine of information. Happy Cruising.
I've just come back from a caribbean cruise and we were there in January - the weather was near perfect, only one day when it was cloudy. There were maybe four or five random showers throughout the fortnight, but these were over and the sky was clear within half an hour. However, we met someone who claimed to have been on a cruise during a hurricane, and they said they were sunbathing on deck during said hurricane...