Your GCFI is being tripped because you have an exposed connection (most likely an extension cord?). Waterproof the connection.
You'll need a foundation masonry bit and a hammer drill to get through the basement, other than that it's hard to screw up low-voltage wiring. Follow the directions on the box.
Those two-prong plugs deteriorate over time with temperature changes/UV exposure. I never use them. You're better off cutting them off and then hardwiring it all together. It doesn't matter, it's all a continuous circuit and you'll have much better wire-to-wire contact with the hardwire method rather than trying to get those flimsy little metal prongs through the main feed wire casing. Electrical tape or heat-shrink tubing seals the connections nicely.
For added protection of the transformer unit, get yourself a good quality plastic box with a lid (hinged or not). Mount the plastic box to your substrate (deck post, fence post, whatever), cut a hole in the plastic box for the power cord and the 12V feed wire, then attach your transformer to the substrate (deck post, fence post, whatever) tinside the plastic box. Put the box cover back on and you've now protected your transformer from the weather.
When buying a 12V transformer, always buy more wattage than you think you'll need. Garden lights are generally around 7W apiece, with underwater lights and landscape uplighting being in the 20-30W range. Add up all the light totals, and then get yourself a transformer that's about 30% more than what you think you need. You won't regret it if you decide to add lights in the future because you'll have the excess capacity. The difference in 12V transformer prices (as far as total wattage goes) isn't a big issue.
As far as the underwater lights go, if water gets into the casing, the lights will still work, but the base of the MR12/16 bulb will deteriorate. Marine sealant is a bit of a pain when it comes to bulb replacement. It's for that reason that the LED lights are the way to go (longer "bulb" life). LEDs also have a much lower heat signature. Power consumption isn't an issue, it's a 12V low-wattage system.
I bought some lights a few years ago that were actual stones that had been drilled out, and had a 20W flood MR16 bulb in them. Can't find them any more, now all I can find are "resin" (read plastic...Yuck).
To convert existing 110V underwater lights (or garden lights for that matter) to low-wattage 12V system is relatively easy. Simply remove the existing wiring, replace with the 12V low-wattage feed wire and replace the bulb housing with 12V low-wattage sockets (you can get them pretty well anywhere). Underwater low-voltage lights don't have to be waterproof, but the bulb will last longer.
