i feel your pain. part of my garage ceiling fell in during hurricane Katrina due to this very issue. i have not fixed all of it yet, but i did manage to stop the moisture issue. the vent by the garage was actually a false one on my house, you could not even see it from inside the attic. i cut out teh plywood from teh main roof to gain access to the back of the vent and then covered the vent with painted plywood to stop water blowing in. so far it has worked well. now i just have to hang a piece of sheetrock and tape the joints, then reblow the ceiling, then repaint.

you can buy covers with smaller holes, but i chose not to due to cosmetics. mine are old style louvers that were custom made and would take away from the appearance if they were covered externally.

Does your garage have an accessible attic? If so, you could install a drip pan that drains through the soffit fascia.

These are made for central air units and also for washing machines. It's basically a large sheet metal pan with a drain hole at one end. A piece of pipe or tubing goes from the drain hole, across the attic, and out through a small hole in the side of the garage or through the fascia and into a gutter.

The pan in my attic (for my central air unit) is large - about 3 feet x 5 feet. If you installed this so it hung just below the cupola, it should catch any rain that gets blown in.