We bought a house with a basement about 5 years ago. The previous owners finished the basement. They insulated the ceiling, then drywalled it. Personally, I wish the ceiling was just a false ceiling instead of drywall, to allow for accessibility if it's ever needed. But the insulation is definitely a sanity-saver with the kids. The family room and television is directly below our bedroom, and without the insulation, it would be unbearable.
There is one section of the basement that is a store room and also contains the furnace and well pump, unfinished, and the ceiling is not insulated. You can definitely tell a difference when you are inside of it, being able to hear much more sound transmission through the floor from upstairs.
I also run a dehumidifier in the basement. I keep it right about 50% humidity. The house is newer, I think about 15 years old now, and from what I can tell (by the unfinished store room), all of the basement walls are insulated, with vapor barrier that was glued along the bottom.
ETA: There are also supply and return vents in all the basement rooms (family room, office, bedroom, and store room), so there is an air exchange going on during both the heating and cooling seasons.