The correct approach would be to "Decouple" your wall if you dont have a staircase on the same wall.
The "Band-aid" approach below is not the best but, will deliver some relief if you cant afford to install a sound control wall system and decouple the wall.
Normally staircases share the same wall for both units and this is why the "Band-aid" approched is used.
Randy,
I'm still have some trouble understanding why It's so loud because the wall is already decoupled. It's a double wall with a 3" airspace. And 2 layers of 5/8 drywall on either side. I can feel the vibration when they run up and down the stairs when I put my hand on the wall. There are no materials that are continuous between the 2 units with the exception of the thin plywood subfloor because they can't have an air gap between the stories between the units. The plywood has 2 saw cuts to decouple it under the base for each part of the double wall. This wall is already supposed to be high 50's STC in the field. I do know that the neighbors stairs are attached to their side of the double wall.
I have the same problem as the other poster in this thread as it is a 3 story townhouse.
I still don't know what to do about the carpet as I will have to remove the molding on the wall, and the carpet tacks and pay someone to put it back together. The cost of the green glue is going to be less than a tenth of the labor if I can find people to do this for me. Then I'm worried after spending thousands it might not work.
Dave