I used to have this problem pretty perpetually. I feel like the case itself may be an issue (at the time I was using a case with poor ventilation) and drilling holes can certainly help. My current case has vent holes built in.
Here are some things that I think contributed to my problems in the past:
1. using the same few reeds all the time, so they never totally dried out
2. using the same reed cup without really cleaning it out between infected reeds
3. living in a more humid climate at the time (Memphis, TN)
4. using a case with poor ventilation
Here's how I solved the problem:
1. Bought new reed case, new reed water cup, fresh stock of cane.
2. Made all new reeds on cleaned equipment
3. Did not store the old reeds or water or case near the new reeds/water/case
4. Once I had two working reeds (enough to get me by) I threw out all the old stuff
I've not had issues with mold since. I sometimes get "slime" on my reeds, but it's not a bacterial or fungus growth, just build-up that shows up when I soak a reed sometimes but easily cleans off and doesn't come back.
Another hint, don't store the reed case inside your bassoon case. The ventilation isn't very good in there, and could be a cause of issue. Keep it in the pocket of your case cover or something. That's also easier to leave the reed case out for a few hours after you practice then slip it in your case cover later (but don't forget it when you go off to rehearsal!)
M.M.A., D.M.A. University of Illinois at Urbana/Champaign: B.Mus. Lawrence University
Bassoon professor at University of Wisconsin Eau-Claire
Maker of the Little-Jake electric bassoon pickup and Weasel bassoon reeds