As you have learned, electrical tape by itself is not a reliable way to waterproof a connection. Depending on your needs, you could use "liquid electrical tape", Plasti-Dip, linerless rubber electrical tape, or a really thick sticky rubbery tape like Vapo-Wrap or
3M Scotch 06147 Vinyl Mastic Moisture Seal Tape.
In any case, put a layer of good quality electrical tape like Scotch 33+ over anything else. Outdoor weatherproofing is not the place to use cheapo drugstore electrical tape.
A really watertight (but non-removable) splice enclosure can be made from an enclosure like a pill bottle or 35mm film can. Run the cable through the bottom, put the splice inside, and then pot it with 5 minute epoxy or maybe RTV silicone.
Another way to waterproof individual wire splices is to use silicone gel-filled phone splices, like
THESE from Rat Shack. These are what Davis sends when you need to replace your anemometer/wind vane. Note: Unless you really only need 4, these are vastly overpriced at the Shack. You can generally get them in boxes of 100 for 12-15 bucks at an electrical supply house or on-line. Look for "UR" connectors.