Definitely not, because there is the acid in vinegar that causes damage the varnish (topcoat) of the hardwood floor and stair. Probably you did not know the damage right away, but by using many times you will see it.
How about the hardwood floor cleaner? It is ok for the floor because it is designed for the floor, but I did not recommend using it on the surface of the staircase. Because the cleaner is using some chemical on this too and they are different finishing topcoat processing on the floor and stair.
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