Side note: I heard the tanker pilot says something like "we have a wet hose". Is this phrase common?
If you're VMC and simply have rain falling, the procedure doesn't change, and it isn't a big deal (talking fixed wing refueling with a boom here; other communities' procedures/capabilities may be different). The receiver's windshield will have some amount of rain hitting it, but if it isn't particularly heavy rain then he should still be able to do the AR.
If the visibility is really bad because of the rain, or if you're getting into IMC, then that's the situation you have, irrespective of rain falling or not, and obviously you'd rather have better conditions for doing the refueling. Going IMC momentarily while you're hooked up may not be a big deal, as you can still see the tanker, but if you lose sight then the first priority is to get positive separation between the two aircraft.
The procedure is no different to if it's not raining unless the rain is heavy enough to impair the visibilty from the receiver aircrafts cockpit.
Air to Air refuelling is normally planned to be carried out above cloud but there may be operational reasons to do it elsewhere
The terms wet and dry refer to whether fuel is flowing or not. A wet contact is one where the receiver receives fuel from the tanker. A dry contact is one where contact is made but no fuel is transferred, this is done for training as the process of making and breaking contact needs to be practiced