Hurricane season runs from June 1 through November 30 ... and until last year, there had been a hurricane in every month but April ... last year there was one in April. The highest likelihood is August and especially September.
I cruise the Emerald Princess, Eastern Caribbean on April 16, 2012
The important thing to remember is that ships really do have sophisticated instruments and reporting mechanisms that will allow them to avoid any serious encounters with hurricanes. No captain will put his ship in harm's way. It just won't happen. If a storm is between you and a scheduled port, you'll go someplace else, and in the Caribbean, you probably won't notice the difference.
Of course, that doesn't mean that you won't encounter some motion to the ocean. But that can happen anywhere, any cruise, hurricane season or not.
Bottom line: hurricane season is not something to obsess over, nor is much of anything else when it comes to cruising.
You can get Bonine at any drugstore. Over-the-counter. We have a bottle that we carry on all cruises. We've yet to crack the seal. I have to check the expiration date--probably need to get a new bottle that we won't use.
The first of October still sees a lot of hurricances. We live on the Gulf Coast and have had some bad ones in October. But - chances a good that there WON'T be one while you are on your cruise. The ship can avoid anything that might be out there.
Whatever you get for seasickness, TRY IT OUT ahead of time. DH didn't react well to Bonine (meclizine) and had a miserable cruise.