I don't like having a door at the entrance to a stairwell. I'd first try enclosing the top of the stairwell with a doorway to see if that helps.

Consider acoustic tiles for the stairway ceiling, and hang some soft art - canvasses with padding taped behind them.

Adding a solid door(s) is a great idea to reduce noise transmission. I did the same at my house. edit - we close the doors only as needed and, with multiple HVAC zones/thermostats, we have no issues with heating/cooling.

Looking at your set-up, I would choose to frame out at the top of the stairs creating a landing of about 4’ x 6’(?). I would have the new wall with the door(s) be in line with the wall with the laundry basket in front of it. The other new perpendicular wall should be in line with the stairwell wall with the railing on it.

Use solid double doors and consider adding a recessed light over the landing in front of the new doors. A double bookcase door like the one below could be fun:

MY first thought is rugs on the first floor you have nothing to absorb sound there . I do not know where the TV is upstairs but I assume in a room so close the door. Closing in that staircase and adding a door will create issues with heating and cooling so IMO no door. You can do panels on the wall inthe TV room to again absorb sound but really closing the door should be the best.

" A double bookcase door like the one below could be fun: "

At the top of the stairs?

“At the top of the stairs?”

Just one possibility. I used the word ”could” in my suggestion as I don’t know the measurement of the landing I proposed. If it was at least 4’ deep and the pivoting bookcases faced into and opened into the upstairs TV room, they might work. Or might not.

It's kind of dangerous to have a door either at the very top or the very bottom at the end of a run of stairs. At the very top, you could open it and fall down the stairs as there is no "landing". At the bottom, you could fall down the stairs and there is no landing.

This is why while in old houses you will often see basement stairs or attic stairs with a door right at the top or right at the bottom but you no longer see this. It is against code if your area actually pays attention to code. You need a landing that is at least three feet in the path of travel at a change in stairs or at the top and bottom before a wall or door in my area which I think is minimum. In other areas it could be more.

In your house this would require an alteration of the attic pulldown as well, it looks like, but you could have a closet there with a door that opened next to the door to the stairs so you could use it. (This is not a permanent stair so there are not the same requirements). That way the door to the stairwell could also open Onto the landing instead of out into the hall.

You all incredibly helpful! So I guess door is probably not the solution. Are any of you familiar with acoustical tiles and where I would place them? I love this bird painting. We have in the hallway and it is so beautiful just there by itself, but if I had to put up acoustical tiles, maybe on the walls and the ceiling? Just not sure where to put them what fabric to use

A sliding pocket door that slides into the wall might be the perfect solution.

Pal, my NY home has a door at the top of a stairway. We have never had any issues.

A door might work, but you could also muffle sound on the stairs with more sound-absorbing stuff. Like a quilt as a wall-hanging.

Amazon sells acoustical tiles you can attach to the wall

I will look into these!

Is there insulation in the bedroom walls? Are the upstairs doors sollid wood? I wonder if instead of addressing the stairs, look at isolating and sealing the rooms upstairs so the noise doesnt travel down, and the noise from downstairs doesnt get throught the bedroom walls.

I would be nervous to have a door at the foot of the stairs. The problem with a pocket door is that it is hollow where the door recesses, which isnt as sound resistant as an insulated wall. You can use a solid pocketdoor and build out the wall -essentialy add an nsulated layer adjacent to the hollow wall. I did this with the pocket door from my masater to my bathroom so that noise wouldnt travel through the hollow space.

The stairs open to a family room that has a tv. It has thick carpet floors.

I wonder about a ceiling mount curtain rod at the top of the stairs with heavy velvet drapes - they could be pulled across when needed and stacked against the wall when not?

I would also try sound machines in bedrooms to help with downstairs noise and headphones can help with TV noise (either the TV viewer can use them or the reader can use them to block noise/listen to something relaxing when TV is on)

Good idea about heavy curtains. That might work.

I like @la_la Girl‘s suggestion of heavy curtains at the top of the stairs. If that helps diminish noise, you can consider whether the cost of building out a landing with solid door at the top of the stairs is worth it if it gains you even more noise reduction or if opening a set back door is easier/safer than curtains.

I had a similar setup where the noise (and sight line) traveled straught up from the living areas into the bedroom/bathroom areas. Adding a landing with solid doors at the bottom of the stairs during a home renovation was well worth it to gain the quiet upstairs!