If you have a poured or block basement, then you need to prevent water egress. Concrete is porous. It is essential to seal the concrete. Then put up the stud frame. Fiber glass insulation is best, placed between the 2 X 4s. Then you place the polythene sheeting over that stapling it to the 2 X 4s. Then you are ready to place the drywall.
Basement floors are a problem, and a source of moisture egress.
The best options are tiling, but not for an AV room. Unfortunately you can not use wool which is best acoustically.
Do not use a fibrous underlay. I recommend a polyurethane underlay, and a synthetic carpet.
Since this is an AV room, you want to put insulation in the ceiling, not only for noise control, which it does somewhat, but for acoustic reasons. Since this is an AV room you should put insulation behind all interior walls.
Run in wall conduit to all locations. Do not put any cables behind any wall or barrier not in conduit. This is an absolute rule.
Another tip is to run AC mains for TV and any sub, back to the equipment location and ground it at one location. This prevents ground loops. Pay real attention to your grounding plan. This pays huge dividends. The whole design should concentrate on avoiding ANY possibility of a voltage between grounds. This includes paying great attention to cable and ethernet installation, which are a potent source of ground loops.
Make sure there is insulation behind all walls and in the ceiling. Then with an adsorbent carpeting no other acoustic treatment should be necessary. Insulation behind the dry wall is all I have found to be necessary. Since this is existing construction, I doubt you are in a position to design with optimal room dimension ratios, but if you are, then do.
In my experience if you follow those rules and have good speakers with uniform dispersion, no other acoustic treatments will be required.
Here is a link to my room build, which you may or may not find helpful.
Here are some more pictures.
I have found though, basements to be a problem. I don't think there is a fool proof way of staying out of trouble.
I have owned four houses with basements, two of which I built. One of those four only had a very partial basement, and yet I did have problems with all of them from time to time. I don't think these is a guaranteed way of avoiding every problem with basements. That is why in this our last house, I made darn sure it did NOT have a basement.