In the US the hurricane season runs from June 1 to November 30. The Atlantic hurricane zone includes the Gulf of Mexico, the Caribbean, and the US Atlantic Coast. The best way to have a successful trip is to get insured, have emergency supplies, and monitor the weather in advance.
When doing this type of traveling it is smart to get insurance because this will cover expenses in the worst case scenario. It will be needed in order to recover the costs associated with the trip in the event of a hurricane. Make sure to read all of the information and ask questions up front to prevent any aggravation and stress after the fact. If a hurricane shows up on the radar half way through the trip and the remainder of the trip needs to be cancelled, it would be good to know that the insurance will cover it. A good travel agent or an experienced traveler can help choose the right insurance policy for the risk.
In advance of the trip and during the trip it is a good idea to keep up with the weather updates. Tropical depressions are formed deep in the Atlantic so there is ample time to monitor and decide how to proceed. Since hurricanes don't instantly form as hurricanes it is wise to keep monitoring the updates frequently to see the projected path and strength of the storm on an hour by hour status. There are many websites that provide accurate, reliable up to the hour updates such as the National Hurricane Center.
Being prepared is about knowing what to expect and planning accordingly. This includes having a first aid kit, water, non-perishable food, a flashlight, and a local map. Consider other items like extra batteries, a radio, and waterproof matches. It can help to read reviews from other people who have traveled to the specific destination and learn from their experiences. Knowing the area and procedures in advance can make an emergency situation manageable. Many people are more than willing to help and answer posted comments on message boards or direct e-mails.
The risk to traveling to any destination in a hurricane zone during the season leads to less travelers and lower rates. This is primarily what lures the few travelers in at such a risky time. When it comes down to it you are the one that has to make the decisions about the trip and has to manage the risks involved. Use all of the resources available including websites, research, travel agents, and forums to get the most out of the trip and stay safe.