Ductwork. There are no registers in the basement, so a large part of the sound problem is noise is getting through the duct walls and bouncing up the ducts/vents and out the registers in rooms. On a quiet night, you can pretty easily listen in on conversations in the basement. Jamming out to records, even with the volume turned down to a respectable nighttime level—might as well have a speaker right inside the registers in each bedroom.
It seemed to me that a large part of the issue could be solved by stopping sounds from getting into the vents, either with some kind of foam/fiberglass insulation batting, possibly followed by an additional layer of tin on the outside. Of course, this is just my "it would probably work" theorizing. I thought by asking here, I'd find out which product or technique to use, and which to avoid. Not so easily done, apparently.
Additionally, the ducts in the basement don't actually go between the joists, they hang just below them. Would be nice if they had been fitted between joists as I think that would make solving the issue that much easier. There are no beds on the first floor above the basement, and with carpeting and what not, I don't think there is so much getting through that way regardless. Also, there is a large "main duct" going longways along the center of the basement which is (roughly, I'm at work now so just guessing) 8-10" tall, maybe 16" wide, and 20ft long or so, which all the smaller ducts come off of and run to their particular sections of the house. Seems like this large rectangular walled duct is more likely to vibrate like a drum, rather than reflect sound like the round ducts would. Again, just armchair theorizing.
That's the thing... heat radiating off the ducts warms the basement fairly well. Doesn't really need vents in the basement. So by solving this problem, I'll be creating a new one. Making my basement colder!