Textiles are your friend!

Consider a large area rug and runner down your stairwell to start. Depending on the intended use of your basement, a variety of furnishings and accessories will help with sound absorption.

Best of luck with your project!

There are two forms of noise. The first is echo. That's when a noise (like talking on the phone) originates in the space and then 'hangs' in the space (bounces from wall to wall to wall to ceiling to windows to flooring again and again and again). Then there is noise 'transfer'. That's when a noise (said talking) moves from one area to another. That can be 'down a hallway' or 'up the stairs'.

To stop noise TRANSFER you need to stop the ECHO. You need to DEADEN the sound at the SOURCE. The space that the noise originates in (aka 'the tv room') needs to be dealt with so that noise does not 'echo' (does not bounce around) and therefore does not TRANSFER.

To 'deaden' the noise at the source you need soft(er) things...like carpet on the floors, drapes covering large windows, noise reducing tiles (the foam-like things) on the ceiling and large stuffy fabrics in the space (like a fabric sofa and fabric arm chairs and fabric bean bag chairs, etc). All of those 'soft' finishes will DEADEN the noise in the 'noise producing' room.

The INSULATION in the walls, under the flooring, in the ceiling, around the pipes/air ducts, etc are the things that STOP noise TRANSFER. Without insulation 'inside' things you can have all the noise deadening products you want but you will still have noise TRANSFER. You need both systems working in tandem to deal with troublesome noises.

Doors stop noise moving up and down a stairwell. Interior solid core doors are the best at doing this. If you are hearing too much noise coming 'up' or going 'down' the stairs, then the fastest trick is to add a solid core door. If noise is going DOWN then you need the door at the TOP of the stairs (closest to the noise source). If the noise is coming UP the stairs then you need the door at the BASE of the stairs (again, it should be closest to the source).

• Cover the Floor
  Carpets and rugs do more than provide soft padding for your feet. They also deaden sound reflection in a room by absorbing echoes and muffling noise. In rooms that have hardwood, tile or another type of hard-surface flooring, place area rugs that also complement your décor. Additionally, you can also add wall-to-wall carpeting. It is more “permanent” and expensive compared to rugs, but it will definitely reduce echo while providing full-room comfort.

• Cover the Walls and Windows
  Wall and window coverings reduce the amount of sound reflecting off window glass and hard wall surfaces. Heavy, lined window curtains and draperies that extend to the floor help muffle sound, reducing echo and ambient noise from inside the house and out. Canvas paintings, tapestries and decorative wall hangings also can absorb sound to reduce echo. Keep in mind that framed pictures and artwork enclosed behind glass will reflect sound much like windowpanes.

• Fill Rooms with Furnishings
  Rooms filled with plush furniture and accessories help cut down on echo. Couches and plump recliners, for example, absorb sound reflection. Fabric upholstery typically absorbs sound more effectively than vinyl or leather. Even items like bookshelves filled with books and decorative items can reduce echo. Essentially, the more objects there are in a room, the less sound will bounce around it.
  Helpful Tips
  Use soft accessories like decorative throw pillows or blankets to complement your furniture, add flavor to the room’s décor and reduce room echoes.
  Because bathrooms and kitchens have many hard, reflective surfaces, such as tile, echo can be reduced by strategically hanging plush decorative items around the room, such as bath towels or tablecloths and fluffy dishtowels.