I have discovered how to build basements in Sims 2.
Use the lower terrain tool to make a big hole in the ground. Level terrain at the bottom so your sim will have a clear walking surface.
Surround the hole with at least one tile's worth of foundation, all the way around.
Add a staircase. The dynamic rather than the preset single-story stairs seem to be best for this.
Use floor tile to cover the area above the giant hole. It won't look like it's possible at first, but you can build on it one row of tiles at a time until you're done.
Use shift-floor tile to cover the ground at the bottom of the hole with your preferred floor tile. The game will then automatically install walls for your new basement, which you can then cover with wallpaper.
Continue building your house upwards as desired.
Next on the agenda: attempting to build a "tree" house. I have elevators now, so perhaps it will be feasible after all.
(No, the mood setting is not a mistake. I started playing Sims to distract myself from how pissed off I am.)