It depends on the regulator. If the first stage is sealed, then the only major path of water into the inner workings of the regulator is through the inlet fitting so ensuring that stays dry and has an o-ring sealed dust cap is important, but a quick rinse will work pretty well to clean any salt off the exterior.
Whether the first stage is sealed or unsealed, with a sealed dust cap in place, you can always rinse the first stage thoroughly in warm water at home later, just avoid soaking unless you are absolutely sure the dust cap is sealing correctly. Trying to inhale through the reg will reveal any leaks - and a leak could be through the first stage dust cap or through the second stage diaphragm or exhaust valve. If you can't suck ANY air through the reg with the dust cap in place, it's safe to soak. If you can draw some air through the regulator it's a clue that their may be an issue with a leaking diaphragm or exhaust valve in the second stage. In some cases it can just be dirt or salt on the lip of an exhaust valve or seat, so rinse the second stage thoroughly and check again. If the problem persists consult a reg tech. In some cases a very slight leak when sucking as hard as you can is "normal", but any significant leak indicates a problem.
When you soak the regulator try to avoid depressing the purge as that will open the valve in the second stage and allow water to enter the LP hose and migrate to the first stage. If the reg has a seat saver feature that leaves the seat off the orifice in the second stage, disable it or if you can't, don't soak the second stage, just rinse it with the second stage hanging by the hose.
A couple post dive checks will help you find problems before the next dive day.
1) While you still have a tank on the boat (assuming you don't own your own tanks) a IP gauge will let you check the IP to ensure the swing is normal (10-20 psi), the lock up is solid (the needle swings up and stops fairly "solidly, rather than gradually slowing to a stop), there is no creep (the needle actually stops and holds at a given pressure with no movement) and the IP is in the correct range (usually 120-145 psi). If you don't dive on a regular basis, check it again as a pre-dive check. Checking IP as part of a post dive or pre-dive check is always a good idea, however, I disagree with a comment above as if you're having IP creep after an overhaul, find a new tech because it is not normal and an reg with creeping IP should not leave the test bench until the problem is resolved and the creeping has stopped.
2) The above mentioned suck test is easy to do by just turning off the valve on the tank and breathing the residual air in the reg down, until there nothing left. At that point the reg is still sealed on the valve and you should get no air through the reg, exhaust valve or diaphragm.
