You still believe she just fell down the stairs lol. This is what really happened:
There was a prophecy of a child living in Shimotsuki village that would one day defeat Mihawk. Once Mihawk heard abiut it he went to that village and paid close attention to the dojo.
When he saw Kuina and how she neg diffed everyone around her, he thought "that's my future foe. Git gud, bitch, I ain't letting you defeat me".
So he dressed up as a common villager and talked to Koushiro, filling his head about how women can't become strong swordsmen because of their weak bodies. "Fool Koushiro, good thing you don't know about Big Meme", thought Dracule to himself.
Koushiro filled Kuina's head with this poison amd thus her confidence was shaken and her guard was down. Mihawk noticed that with his Hawk Eyes (DAE Clairvoyant) and pushed her down the stairs, killing her instantly.