What is the most cost-effective way to reduce sound coming up my staircase?

Dampening works to help eliminate the reverberations; the sound waves making other solid hard surfaces re-transmit the waves. Sound waves themselves travels in air. If the staircase is open to the other living areas, sound dampening on the walls will do little. Block the air and you block the sound. Can you put a door on one end or the other?

Have you tried a white noise machine? The problem is our brains have evolved to listen to noise to protect against predators and even if you reduce the sound volume you will just end up straining to hear the words. A white noise machine (plays an endless loop of ocean waves, or rain, or some such) gives your brain something to pay attention to. Since it is closer your brain assumes it's more important to focus on the closer sounds from the closer predators than the far away sounds

Otherwise if you can't door it off, and you don't want to go the white noise route, use noise cancelling headphones and listen to soft background music. Those will work as long as they can seal against your ears.

The "most efficient" way would probably be best determined by a full acoustical analysis of your house. Since that would take time and money you probably don't want to spend...

As a way to temporarily test it out:

• lay a 2x4 flat on the upstairs floor, along the bottom of the railing.
• Use string to attach the panels to the 2x4 and hang them along the walls.
• Move them around until you find an arrangement that seems to provide the best noise attenuation.
• Permanently attach panels at these locations.

Since those panels are a bit... unattractive (unless you're into the whole "recording studio" vibe), consider hanging paintings, artwork and tapestries on the walls of the stairwell. Anything that has depth (paintings with chunky frames, artwork), or softness (the tapestries) will interfere with the reflection of sound waves coming up the stairs and help attenuate the sounds. A thick carpet on the stairs will help, too.

As noted in other answers, though, your best bet is going to be blocking the sounds from getting into this space in the first place. Since you have an open staircase that you can't install a permanent door on, consider a temporary door made of heavy fabric (like a blanket) that can be hung from upstairs (consider the 2x4 on the floor trick again), dropped down when you're working, and raised back up to regain the look when not needed.

As you've got a layout that doesn't lend itself to exactly blocking the sound, you'll need to look into several layers of attenuation.

The first is close to the source - blocking the direct line from the TV to the bottom of the stairs. Upholstered wooden panels are good, like office cubicle screens (though hopefully nicer looking), or bookshelves with backs. As well as reducing the sound heading towards the stairs, this may also allow the person watching (or meeting online or whatever) to turn the volume down a little by blocking sounds they can hear. If it's a conversation, it may be quieter.

FreeMan's answer covers the stairwell itself (step 2) better than I could.

Step 3 is around your desk. A panel between the desk and stairwell will help a lot. Again, suitable DIY-able material would bewood with an upholstered cover or perhaps covered in cork tiles as a pinboard. This should also extend round the sides of the desk, where closed-back shelving may be useful.

All these should extend as close to the ceiling as possible. Soft stuff is much more sound-absorbing than hard, so carpets and curtains/wall-hangings are all good.

When it comes to watching videos, the multiple speakers of a surround-sound system complicate matters. They tend to be used louder, and to deliver more bass, that travels through the house. Isolating any speakers from the floor and walls will help, especially subwoofers. On the other hand a well-tuned system can lend itself to being turned down a little - you may want to adjust settings depending on the type of noise that comes through strongest.