OK Brooke,

It just so happens that I am not only a contractor, but an accomplished musician and sound proofing has been an area of study for me.

The best solution for this situation is to remove the door and frame it in so that drywall can be hung on both sides to properly finish the wall.

Before hanging the drywall though, fill it with insulation and you just finish the drywall and blend it in with the existing wall. This is the best solution for moderate results, but since you might not want to go to all that trouble (or the landlord might not want to) here's another solution.

One of the best sound barriers is rubber underlayment. You can find it at a tile store or tile supply warehouse. I believe some of your big outfits like Lowe's even carry it. It is basically a thick black rubber sheet. If you are looking for a temporary fix, you can cut it to fit around the door. You need to overlap the door's trim by several inches and the trick is, it has to be completely sealed. You can do this with tape (if you're not looking for a nice looking finish. The truth here is, aside from removing the door as I mentioned above, there isn't a 'pretty' way to fix this. Weather stripping won't keep the sound out that well, but getting any gaps or holes filled will help a bit.

You can try replacing the door with a solid core door.

I hope some of that is helpful. Let me know if you have any other questions.