first, remove the tile panels in the area you want the door header and take a look at the support system. Take a photo and measure for comments.

Build a header and install a door BTW why do you want the door ? The pics do not help at all BTW

It would be a terrible design to construct a door frame and have a door close to that bottom step. To do this right, and not have an ugly, impractical result:

1. Remove the suspended ceiling from the area between the stairs and where the door will be.

2. Build a wall out about 2' long starting where that diagonal wall paneling is.

3. Build the door frame connecting the end of this new wall to the corner of the existing wall (the corner where the baby gate is leaning against).

That way you will have a bit of a landing when you come down the steps. If you don't do this, and have the door closer to the bottom of the steps, it will be awkward going through the door, and taller people will have to duck.

I don't know if you have the skills to do this, and do it right, but there is no simple way to slap a doorframe and door in that space and expect it will be a good solution.

Life safety code requires a landing at the bottom of the stair, and that should be 36" deep the length of travel, so your plans for a door do not work. Also, you can verify there is min 6'-8" from the stair tread to the bottom of the suspended ceiling.