I recently saw a commercial for that garage floor finishing stuff. It
>looks really good, and I'd like to do that with the unfinished part of our
>basement. Since no cars have (to my knowledge) been parked in my
>basement, the floor down there is pretty stain-free and it's never been
>painted. Would I need to do any serious scrubbing/cleaning work on it and
>prime it first?
"Probably", unless it's approximating new construction -- in which case, you've got 'green concrete' issues to deal with, instead.
A medium-potent detergent solution, scrubbing with a stiff bristle 'scrub brush' (the objective being to loosen any 'ground in' crud), followed by a *thorough* rinsing. After all this, you'll need to let it dry thoroughly, before applying the 'finish' -- absolute minimum a couple of days.
As for priming first, "read and follow label directions" for the "finish" product you intend to use applies.