There is no detectable moisture that comes through the walls or the
floors.
I tested this using the "tape and seal a piece of aluminum foil" trick on
the walls and the floor. After a few days, when the taped and sealed foil
is removed there is no moisture there. Despite all of this, there is a
VERY
SLIGHT musty smell in the basement -- it is almost undetectable, but it's
there.

Check for a basement "floor drain"..often nearby the water heater. I think
they' re mandated by code so there should be one there somewhere.
There's supposed to be a trap in these drains but they need water inside the
trap to be effective. (Some are installed with a small waterline to keep
them "filled" but most are not). See (smell) if the odor is coming from the
drain. Take the screen/grille off. It may need a cleaning in the event its
been left unattended/neglected with crud in it for years as is usually the
case. It may need a rinsing and refill of the trap..water with a little
bleach has worked for me.

Otherwise, I like the idea of putting return air vents in the basement. Our
house was built with them. I don't think much of builders who don't put
them in when finishing a home although many simply rely on the one BIG ONE,
attached directly to the Air handler, especially if the basement isn't going
to be "finished" right away. The reason for this being that they don't know
where the "next guy" may be putting walls/rooms etc when the basement does
get finished. Give some thought to that and let the HVAC guy know where
YOU'd like the vents (think..where might I put bedrooms/bath etc. later ? )
as opposed to where the AC guy "thinks" they should go.