Should basement staircase door have a passage or privacy knob?
You can install either by code....in fact you don’t even need a door.
However, a landing is required at the top of those stairs, if the door swings outward over the stairs. (See ICC R311.5.4) If that’s the case and no landing is provided, I’d put a “keyed lock set” on that door so no one accidentally uses the door, until you can fix the door swing or just remove the door. (No door...no landing required.)