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.