Here is the problem: The pipe from the first floor bathroom sink
basin leaks. We know this because, fortunately, the basement ceiling
is not yet finished, and we can look right up at where the black ABS
pipe comes down from the first floor and we see the water dripping
down the pipe. Behind the bathroom sink, the pipe goes horizontally
into the wall, then must turn down 90 degrees and then presumably does
not turn again until we can see it above us when standing in the
basement. When in the basement, we see the water leaking down the
pipe, but since the pipe disappears into the framing above, we can't
see the source. Therefore, I am guessing the only reasonable source
of the leak is in the wall where the pipe makes its 90 degree turn
downward. Does this sound reasonable so far?

Sounds like a leaky joint between pieces of pipe. This could be at any
point above where the water is dripping of course.

The first thing I'd do is check the drain to make sure it's not the origin
point of the leak. Then I'd go ahead open up the wall behind the sink where
the elbow joint is located and give that a look. At that point of course
you can also check the pipe itself for any issues.

If it's the elbow that's bad, it may be as simple as the piece was fitted
but never glued into place. Be that case, just finish the job. If any of
the pipe is bad then you'll just have to cut it out and replace it. The
nice thing about these pipes is how easy it is to do the work -- they just
glue right together, takes but a moment.

Repairing the drywall shouldn't be that difficult afterwards either -- you
may find studs are right handy and you can just cut back to the studs then
replace the piece you took out. For example, when we rebuilt the wall
between the bathroom and bedroom, it just so happened that the last bit of
area to cover was only a couple of feet wide so it got a partial cut that
covered the plumbing to the tub. If I had to get to the plumbing, I'd cut
the tape, cut an area large enough to work in, then afterward just cut a
piece to fit back in and re-finish. If you're not that fortunate, you can
also use some 1 by braces to hold another sheet of drywall in place then
mount and finish.

James