We were there when Hurricane Francis was projected to hit in September of 1999. It was the first time that Disney had closed their parks up until that time. All theme parks, water parks and Downtown Disney were closed. The only places open were the hotels. Disney moved everyone out of Fort Wilderness and into hotels for safety.
The hotel staff handed out flashlights and instructions (stay away from windows, keep curtain closed, fill tub with fresh water, stay indoors) to all guests. We stocked up on supplies (snacks, soda, beer, water) from the hotel store - we were staying at Port Orleans Riverside. Disney discounted food/drink prices in store by 20 to 30%. The food court and restuarant were open with more discounts - and a very long wait.
We deceided to get a pizza from the food court and hang out in the lounge. Hotel employees were very helpful despite the fact that it was insanely busy and I am sure they were worried about their own homes while they were away at work.
The hurricane made a turn to the north at the last minute leaving Florida with passing showers and wind. The only park to open the next day was Animal Kingdom - employees had to report to care for the animlas anyway. The park was open only to guests staying in Disney hotels. The park opened slowly and in stages as employees were able to report into work. Also Downtown Disney reopened that day as well.
By the following day all of Disney World was operating as normal.
I would not hesitate to make the trip. Disney handled the situation very professionally. At no time did we feel unsafe. In fact, we made our trip last year in September of last year between the hurricanes and would do so again.