In order from lowest to highest cost (roughly):
1) For something simple like this, grease filled wire nuts aren't the most beautiful option, but you can readily pick up a pack of them for a few bucks at your local hardware store.
Pros: Cheap, available, work
Cons: Messy if you need to disassemble, clunky, look bad
2) You could also use grease-filled IDC (insulation displacement) connectors.
Pros: Cheap, available
Cons: Like other IDC connectors, not great at high current, messy to service
3) Use adhesive-lined heatshrink tubing. It works just like heatshrink, but has an inner liner of hot melt glue which will melt as the tube shrinks, creating a (mostly) environmentally protected connection.
Pros: Moderately easy to service, fun to shrink, looks clean
Cons: More expensive, is stiff when finished (can cause stress risers in the wire)
4) Use adhesive-lined crimp splices or adhesive-lined solder splices. Outer sleeve is similar to adhesive-lined heashrink but is usually transparent for inspection. Crimp style are crimped, then shrunk. Solder style have low temp solder pre-applied inside, just insert the wire, apply heat, and you've got a waterproof solder splice.
Pros: Professional, adhesive layer can be inspected to ensure full coverage
Cons: Really, really expensive.
The best existing solution is something that is called parylene, if you can put a film of this down without pin holes then you can have a water barrier and dielectric barrier. Often used in space and extreme environments. I've seen a 20KV voltage source that looked like a bare board that you can hold in your hand.
Parylene is relatively expensive, partly because of raw material cost, partly because of application technique. It is applied as an evaporative coating, which can be very wasteful as it coats the inside of the chamber so only a small percentage ends up on the board.
Whether of not this Liquipel can be used in electronics will have to be tested. What is important is that you form a physical barrier that bonds to the surface. The material that I saw demonstations of in the past (80% that it is this Liquipel) used a nano-layer to entrain/entrap air as a buffering layer. This material will wear off eventually and counts on a rough surface (nanoscopically) to entrap air which then form the "barrier". So it is using van der waals forces and not covalent bonds. These bonds could be saturated with other materials after mechanical abrasion/contact. Also at corners, say the edges of leads on a IC package I could see that this would be a place where this material might fail as a nanoscale structure might not conform around a corner fully.
But these are areas to look at, not necessarily a reason to reject it until proven not to work.
Best Answer
Ask you FR4 supplier about conformal coating options. They can mask it off, on the windows of the LED's and such. Check the water resistance of the LED's and such exposed parts. Most electronics are, as they are washed. Conformal Coating is very common on military and automotive electronics. A long time ago, I used it on video equipment that went in the trunk of vehicles to protect from salt build up from DEW point condensation.