I had this problem for a long time and nobody I took my car to could fix it. However, the solution turned out to be quite simple.
1. There is an area under your hood just below your windshield. This area has 2 drainage flaps on the firewall that allow water to drain out through the engine compartment. Remove all leaves from this compartment and the drains. With a hose let water flow into the compartment and make sure it is coming out of the drains on both sides. If this area is clogged, water will flow into your blower housing.
2. Remove your blower housing. To do this, unplug all wires connected to the housing. Go around the outside of the housing and remove all nuts and bolts holding the housing to the firewall. On my '94 Cavalier, there are two nuts on the bottom which are connected to bolts permanently attached to the firewall. One of these is behind the steering gear and is hard to access. I had to remove both steering pump lines from the steering gear then I unbolted the gear from the mount on that side. This enabled me to pry the steering gear away from the nut I needed to remove. I ended up removing the nut with a universal joint socket.
3. Now that your blower housing is removed you should have a big pile of leaves that came out and there probably will still be come caked onto your evaporator core. Remove all of these leaves and hose everything down completely.
4. Reassemble everthing. You will notice a gasket on the blower housing that goes around the evaporator core. I used a gasket in a can type of stuff. It worked very well.
Run your air conditioning on high and you should see water drip out from under the car.
I hope this helps you.