'Staircase' is literally 'a climbing-enclosure' — the same root gave us 'stile' and 'stirrup.'
A compound of 'stair' + 'case' (enclosure, container). 'Stair' from Old English 'stǣger' (a stair, a flight of steps, a ladder), from Proto-Germanic *staigri- (something to climb, a climbing structure), from PIE *steygʰ- (to stride, to step, to climb, to mount). The same PIE root produced Greek 'steíkhein' (to walk in rows, to go in steps), Gothic 'steigan' (to climb), German 'steigen' (to climb, to ascend), and Sanskrit 'stighnoti' (he climbs). 'Case' in the compound derives from Old French 'casse,' from