If it's just a tropical depression or tropical storm, it's pretty much business as usual - just very wet! If an actual hurricane is approaching Florida, they will determine if it's close enough and strong enough to require the parks to close. They try to allow enough time for CM's to get home before the storm arrives. There will also be fewer CM's scheduled to work at the resorts - they did an extra towel delivery from housekeeping prior to the storm, and advised that you stock up on snacks and drinks for you room because the restaurants would not be operating. They ran extra entertainment on the TV and in some of the resorts that are interconnected they had activities in the lobby. If you are in a resort with exterior corridors or separate buildings, you may be confined to your room for as long as 24-36 hours, depending on how fast the storm moves and how close it is to Orlando. They were very good about leaving messages on the room phone and under the door about what was going on.
I have a 10 day stay booked in September, and I visit in September pretty much every year. I pack a couple good books, a small flashlight, a small weather radio, and plan to have something like bread and peanut butter that doesn't require electricity to prepare for eating. Make sure your cell phone is charged before a storm arrives, as well as any other electronics you have with you. I know a lot of people like to unplug from the real world while at Disney, but it's a good idea to check in with the Weather Channel or the local news once a day just to catch the weather forecasts and make sure everything is clear. I usually put the Weather Channel on while I'm getting ready to head to the parks, so I keep up with it that way.
One last thing - travel insurance. A lot of people ask if they should buy travel insurance during hurricane season, and the answer is it depends. If you're driving, then I would say no, because Disney is very flexible when it comes to cancelling or rescheduling because of a hurricane here. If you're flying on tickets you can't change without paying out a lot of money, then it becomes more of an issue. You also need to consider whether you could absorb the costs if you are delayed leaving Florida because of a hurricane. Can you afford to pay for an extra day or two at a hotel on top of paying for airline ticket changes? The more money you can possibly lose, the more valuable the insurance can be. But read the fine print carefully to be sure you will be covered in the way you need to be covered. Every policy is different, and not all of them will cover additional expenses and cancellations due to hurricanes.
Think positive! I've only had a couple September Disney vacations affected by hurricanes, and the weather just before and after the storm is usually fantastic! I still had great trips, and I've had dozens of trips with no tropical weather at all.
I have been there during labor day week before but had no trouble except some pretty hard down pours. I just wondered about the hurricanes because I was searching the internet and read on a few articles that said it was suppose to be a extremely active hurricane season this year. I hope they are wrong and everyone going Aug & Sept has a wonderful hurricane free trip ! 
