Need better pictures that actually expand and get bigger as well as a floor plan for the basement area with dimensions if you could draw it out on graph paper.

Not possible, or safe, if it were possible. A 3x3 landing is code for a reason. Make some baby gate "cozies" with old quilts is about the best you can do for a temporary noise buffering.

Will a door at the left of the stairs (looking up from the bottom) work ?

No, there has to be a 3 foot long landing at the bottom of a staircase, the door can't be against the steps. There are lots of those pre-existing, but they are not longer permitted.

Might be simpler to get a white noise machine for the baby, assuming that you are concerned about the baby's exposure to household noise, not the other way around.

FYI babies do not need silent house in fact most function better when sleep in not silent but has regular house noises that they need to get used to . I used to vacuum when our kids slept and all of them had good sleep habits because every little noise did not wake them up. As mantioned no you cannot put a door there it is not to code . Make sure the baby's bedroom door is solid and rugs on the floor to keep sound from travelling up and as said maybe a white noise machine could do the trick

Congratulations on the upcoming baby! I tend to agree with Patricia. Babies don't mind a little noise. A textile wall hanging on the wall at that first turn might be all you need, if you even need that.

No need to mess up your house for a temporary thing that probably won't even actually be a problem!

I understand how the open stair well can create a problem as far as the acoustics.

Like others have said, something on the walls to absorb sound will help.

I would also suggest a heavy insulated type drape hung on a pole in the place you wanted to install a door. It can even puddle on the floor some.

Adults and kids will get used to it quickly, and it will be easy to remove when you no longer need it.

Babies grow up fast!

Trying to isolate every household sound from a baby is a classic first time parent mistake! I did it too! Life was miserable and you definitely don't make that mistake with child #2 or any more that might come along afterwards.

Hang a quilt/drape on a tension rod if you must block some of the sound.

If you must add a door and there is sufficient space, consider framing in a door at the top of the stairs. Replace the adjacent railing with a glass wall or a half wall with glass on top.

Isn't there a door on the baby's room?

Please get a sound machine and try not to stress this. My son was a finicky sleeper in a small, one-story house. The sound machine saved us. He’s a teenager now and still sleeps with one for comfort.

My son was one of 22 babies in the hospital nursery. He could sleep! We played music or watched tv at normal volume. We had parties. We talked in the hallway. This noise problem might not be a problem for your little one at all. The second and third and fourth will have to learn to sleep with noise.

Are you trying to shield the baby from noise or shield a workspace or another area from the baby noise?

(just asking because my Father in law used to work nights and slept during the day and needed a quiet space away from kid noise.)

As others have said, babies will sleep through noise and won't wake up with every noise as they get older if they are conditioned from birth to sleep with noise.

If you need a quiet workspace there may be other solutions.