I worked backstage for The Tempest in a 2-inch deep pool (which was always filled) and a rain sequence. That was fun. And I'm sure it involved a lot more water than yours will.
We made a deal with a neighboring Hilton hotel (we shared a wall!) so they gave us clean towels every night, which we returned for cleaning, in return for credit in the program. We had 12 performers and 2 backstage people, and we literally went through 40 towels a night (so we got 40 each night, and asked for an extra 10 to keep around if we ever needed more). If you can manage this, great! Just make sure you introduce yourself and your staff to all the managers/bosses, even the ones who'll seem irrelevant, and tell them all how long the run is. When I or the other backstage person went to get towels the first two nights, everyone was great, but as the month went on they started giving us problems, saying they'd never met us, and asking if we were even telling the truth about the show.
We absolutely had space heaters in the green room, towels over all the couches, and halfway through the show we hung some of those outdoor heater things backstage, so the actors would also be warm while waiting for their entrances. We hung some heat-refecting panels right above the heaters, so all the heat would go toward the actors rather than toward the fire detectors. And each actor had a bath robe they put on the second they got offstage.
Like cdavisnyc, we took one of the backstage rooms and turned it into a drying/wardrobe room, with plastic bins everywhere so we could hang costumes and shoes above them and they'd catch the water.
If the costume designer can treat the costumes, see if they can be made water-repellent. We used Scotchguard on the outside of the clothes (not on the inside so washing them wouldn't be an impossible feat) and water basically bounced off the costumes (like an umbrella). Yay!
Also, our post-show involved literally vacuum-cleaning the set, the floors backstage, under the stage, and the (carpeted) house floors. We used those orange shop-vacs, and had to empty them three or four times every night. We laid some nonslip rugs (the black things every theatre seems to have backstage) backstage, so both the rugs and the floor below got vacuumed. We hung the rugs up on flats to speed drying up, and then left industrial fans, pointed to cover as much of the floor (and rugs) backstage as possible, on all night. I know it sounds like overkill, but we still found water everywhere when we came in in the morning. This at least seemed to keep the mildew down.
Regarding rain - our set-up was "simply" two PVC pipes in the air, with holes all over, connected to our utility sink backstage. We tested it every day and left the water on all show long, with a key right offstage that we'd turn to turn it on. If the decision had been up to me, I'd have done it differently - turning the water on with a warning, 2 or 3 minutes before rain, and turning the key on a GO. Our rain was always cold because the hot water ran out by the time we got to the scene! I insist on testing it every day, though. A couple of days we came in and although nobody had touched the hoses and pipes, we found some connections would leak. I seem to recall that one day our PM even had to go on an emergency run to get a new hose, because ours wouldn't stop leaking.
I wasn't dealing with any electronics and I never saw the show from the house side, so I don't know the actual specifics - but I know toward the show we got a couple of ghosting lights and our projections became finicky. One of our speakers also bit it toward the end of the show. The official diagnosis was that all the humidity in the air throughout an entire month got to the electronics. I haven't worked in that theatre again, so I'm not sure if the electronics got fixed or if they had to replace any of them.