The silicone grease that commes in the kit (the white grease) is water proof (as are most petrol based lubricants; silicone lube isn't petrol based though). If you're not happy witht he run-of-the-mill silicone grease that comes with the kit, you can get yourself some Molykote 111 (comes in a tube). This is really a bit stickier but the dang thing doesn't come off your fingers with anything (not water soluble (i.e. waterproof), not petrol based product soluble).
As suggested above the trick is to make sure it doesn't come out of the diff.
I make my own covers from Avery identification label paper (part #936067-durable heavy duty) which is quite expensive but you get the supply of a lifetime in a pack, or you can on-sell the extra sheets. This is plastic (vynil?) "paper" and has a very good adhesive back. I cut the inside circle a bit small too, so it rubs on the outdrive to seal better. See the picture. With these I basically never have to worry about anything getting in or coming out of the diff.
Team Greasy Weasel
The best upgrade to any car is some driver skill.