Waterproofing outdoor wiring
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.