It seems like your staircase area isn't enclosed ? It's open ?In my case the stair crawl space area is on the ground floor, I did think of putting the batteries in the extreme corner of that crawl space under the stairs but doing so will make it difficult to do maintenance on batteries, In that crawl space try breathing during the maintenance with battery gases, dust and that area is more dustier then other areas, being the stair case peoples shoes carry the dust, dust in the mat, when ever one opens the door near the stair case more dust enters and the water spill above the batteries(it happens during refill) and petroleum jelly makes it a dust magnet.
Mine has a thin plywood sheet that covers it and there is even a door. The door is usually closed and locked. There is a door next to it that is usually kept open as it leads to the top floor. Dust comes in this way. But i think the area under the stairs gets less.