IMO it will depend how much of an audiophile you are . We have a basement media space it is open to the rest of the main entertainment area since we have a vintage pool table ans a bar in the space. We mounted speakers it the media space and have found it works very well to keep most of the sound in that area. I hve never really found soundproofing that works perfectly with this set up but we have only a guest room above and so rarely and issue since when we have guests we don’t usully watch TV after they go to bed. I agree closing that wall to the stairs would not be attractive and like I said it really depends on your level of enjoyment. My husband is much more concerned with the sound from the stereo upstairs than the TV

We finished our basement 2 years ago and I think the audio system we have is Sonos. We did not add any additional insulation to the ceiling. We do not have a guest bedroom in the basement. The area above our tv area on the main level is the kitchen. I have 2 teenagers who have been watching more movies due to covid and sheltering in place, and the only time I can really hear the audio is when I walk past the door to the basement, even when it's closed. Depending on the type of movie they're watching (they tend to watch action flicks) the sound can feel like it's right in the next room. We left our stairs open too because we we don't get much light down there and didn't want it to feel completely dungeon-like.

I guess the point of my rambling is the sound has not been an issue for us. The only time it was was a couple of times over the summer when my son happened to stay up late. We have a 2 story foyer and we could actually hear some of the movie on the second floor. But only big rumbling sounds where the stereo surround was really pronounced. We couldn't hear dialogue.

If i will be open to the rest of the basement soundproofing the ceiling will not do much good. Same goes for a projector set-up. Projectors, to get the most from them, need to be in a light controlled room. Even small amounts of light can wash out the image. If you need the soundproofing then a room within a room is the best way to achieve that but even then bass will come through to the rest of the house. May I recommend a forum where you can get excellent advice regarding soundproofing (and all things home theater related); AVS Forum. I joined several years back when building my home theater and they were an indispensable trove of knowledge! With their help I did this in my basement;