Consider each empty (uninsulated) stud bay as a resonant chamber. Stuffing the bays will change the resonant frequency and amplitude, and more than likely sound better. It will also help to isolate the room from the rest of the house. Another trick would be to install crippler studs / fire blocks in the wall cavity. That also reduces the size of the chambers and stiffens the wall. Stagger the blocks so that you don't have uniform sized chambers.
As far as how much insulation, I'd overstuff the bays. If you have standard 2x4 walls, use 6 inch (R19) batts (or more if you are willing to splurge.) The compressed fiberglass will act as a bass trap.
Heavier drywall is also a good idea. Once again it makes for a stiffer wall, thereby raising the resonant frequency. In this case, mass is your friend.
Ultimately, while you probaby won't eliminate bass issues, you will reduce their effects.