We sailed during hurricane season last Sept. - in fact, 3 hurricanes were spinning around in the Atlantic & the Caribbean while we did. No problems. We weren't anywhere near where they were, so our itinerary didn't need to be changed, and the seas were actually fairly calm - more so than I expected. The weather was hot & muggy, which might be normal for that time of year in the Caribbean.
In short, I wouldn't worry about it. If hurricanes approach, the ship's itinerary will be changed to avoid danger. If they get close, you will of course encounter rainy weather and probably some rough seas, but there are ways to deal with that too. Just go and have a good time!