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.