Probably far more important to cover them than to insulate behind the cover . What are your intentions ?
I'd insulate them in addition to what the coverings previously mentioned. I agree with you also about the ceiling and since the insulators will already be out there to do the ceiling, the walls shouldn't be much more of an expense.
For drywall, I'd use the drywall used in showers to preclude any moisture invasion or damage.
Even though your basement is dry I still suggest the mold resistant (Fiberglass instead of paper) sheet rock available at Home Depot. It meets all code for both regular sheetrock and for green sheetrock used in a shower.
Yes, you should insulate as noise does transfer upward, inside the walls, too.
Also, there are acoustic caulking tubes available and you should use this perhaps at all corners and certainly at base of sheetrock, behind the baseboard.
If that is your intention ie. insulation and a covering , then use rubber strips or a non hardening caulk (silicone ?) between the studs and the sheetrock no matter what type of sheetrock , vapor barrier and insulation that you use . The object is to stop the sound transmission through the material . This is especially important in the ceiling with your application .
I would suggest using styrofoam insulation . You can get various sizes and thickness's at Home Depot , it won't hold water like fiberglass or some other forms of insulation , it is a denser material that works better for sound transmission abatement and you get a higher insulating value per inch .
Insulation cuts heat loss but does little for blocking sound. You should check local code and pracice for insulation in the walls. If you want to cut down sound, then you should look at doing the ceiling with a double layer of drywall with Green Glue in between the layers. You might also want to use resilient sound isolation clips and things like rubber joint isolators and soundproofing caulk. Also, keep the cuts into the ceiling to a minimum (like eliminating recessed lighting). The ceiling is the most important, but flanking sound up the walls can also be a problem. Really depends on who quiet you want it. The are good articles on these topics on the Green Glue website and on the website for the SoundProofing Company (no affliation with either). These techniques can really makes a difference.