No problem! I've cruised twice during hurricane season without difficulty. You just have to know what you're getting into. Here's the good and the bad:
Prices are considerably lower during hurricane season (which, by the way, makes up a full 25% of the year!).
Cruise ships always sail -- they just don't always sail to the places where they're scheduled to sail. If there's a hurricane down around Jamaica, and it'd be dangerous to go there, the captain is authorized to make changes to the itinerary. The ships (and the weather service) are very precise in their hurricane-tracking, so you'll never be in any danger. So you might have an extra day at sea (far from any rain), or you might end up visiting a "surprise" island.
The ship is MOBILE. You will not be taken anywhere near a hurricane, so the water will not be more rough during this season. Large cruise ships have stabilizers, and you're not likely to feel more movement at any particular time of year. If you're very worried about seasickness, book a mid-ship cabin on a low deck; you'll feel less movement there than at other points on the ship.
The weather during hurricane season is ideal: warm enough to be comfortable in shorts during the day, but not so hot that you'll sweat; chilly enough that you'll need a jacket after dark.
Occasionally the hurricane might affect the port city from which you're scheduled to sail -- it could change your cruise dates. If you're scheduled to sail out on Sunday, but the ship cannot dock, the people onboard may get an extra day (or two!) of cruising, while you may end up boarding on a Monday or Tuesday. When you're making your reservations months ahead, there is no way to predict this, and there's no good answer to the problem, if it should arise. However, it is very, very rare.
Most people fly to their cruise port city. It's possible that a hurricane could disrupt air traffic, and you could have difficulty getting to your port (or getting home). Again, this is rare. Also, it's not going to be dangerous; it's just a matter of whether you can get to the ship.
If you're going to cruise during hurricane season, you should get insurance. If you should be unable to reach the ship because of weather, you would receive a refund for your cruise. Also, if you were to find yourself stuck in your port city because of weather, your insurance would pay your hotel bill. It would be disappointing, but you could re-book for a later date.
Yes, it IS a risk to cruise during hurricane season, but it's our favorite time of year to cruise! Just look into all the details and decide whether the risks are worthwhile for your circumstances.