Personally, we like to travel during hurricane season because we get really great rates, plus everyone is back to work and back to school, so usually the ships aren't as full and there are hardly any kids onboard.
While we've cruised in September several times, we've been lucky and have only encountered a hurricane one time. We were doing our first port of call, Princess Cays, during an Eastern Caribbean itinerary, when the Captain announced there was a hurricane coming up from the south and we would have to alter our itinerary and do a Western Caribbean itinerary instead. We had a wonderful cruise with great weather the entire time.
A few days later I was talking to the Captain and he mentioned the funny thing was that all the other ships did the same thing as us with the exception of Carnival. He said he really felt sorry for those passengers because they had one hell of a ride.
During more than 11 years of selling cruises, I've heard that exact same thing has happened on several occasions.
Most cruise lines will not put you in danger in such a matter for the safety of their passengers and the safety of their ships.
But hurricane season is definitely one time of the year when I really suggest purchasing travel insurance. You hope you don't need it, but it's better to have it and not need it, than need it and not have it.