I went to DQ for two consecutive nights back in April. The first was on a Friday in a torrential downpour (the Friday after Easter). This was peak season at Disney so I wasn't sure what to expect. Since we got there at 10pm it was definitely busy but not that bad.
The next night we returned. I was braced for the worst. A Saturday night! Beautiful, weatherwise. We got there about the same time and the place was practically deserted.
So, yes, my very unscientific test points to a very busy DQ on a rainy day. However, realize that the place draws its largest crowd in the afternoon. If you are there when it opens or show up late in the evening, it should be fine rain or shine.
Rainy day suggestions? If you've got a car, Orlando Science Center is one of the best science museums and it's a covered garage and covered walkway into the museum.
Also, it has to be said, if you can hit the park on a rainy day the crowds will be a lot smaller than on a perfect day. Just head out to the parks like Magic Kingdom and Epcot and MGM with plenty of covered attractions (avoid AK). You'll be exposed to the weather between rides, but you have places like Epcot where you can kill a lot of time at Innoventions and Spaceship Earth and Living Seas.