Guidance from published adventures suggests 1d6 per 10 feet, and its up to the DM how it works.
As noted below, the RotF and RoT have slightly differing mechanics. One has the falling creature only fall 10 feet, despite the staircase being much longer, and the other has the creature making a check at a certain point on the staircase, falling 60 feet to the bottom of the stairs on a failure. These are unique features of these individual staircases. It will be up to the DM to determine exactly how a creature falls down a particular staircase, but 1d6 damage per 10 feet of stairs is consistently expressed in multiple publications.
Tales from the Yawning Portal suggests 1d6 for a fall down stairs.
There is a certain room in a certain dungeon with stairs leading up to a central dais. If you choose the wrong stair case, the following happens before you reach the top:
The eastern stairs are shrouded in dim light, which can’t be made brighter by any means. Any character who mounts the stairs falls down and rolls back onto the floor, taking 3 (1d6) bludgeoning damage.
I recently fell down a flight of stairs, and it wasn't particularly injurious, so 1d6 seems right to me, experientially.
Rime of the Frostmaiden suggests 1d6 for 10 foot fall down stairs.
In a certain location in Icewind Dale, we have an icy staircase:
Stairs Down. The stairs, which descend 90 feet to area G10, are slippery. The slippery ice is difficult terrain. When a creature moves on these stairs for the first time on a turn, it must succeed on a DC 10 Dexterity (Acrobatics) check or fall prone. If the check fails by 5 or more, the creature also tumbles 10 feet down the stairs and takes 3 (1d6) bludgeoning damage.
So this one gives 1d6 per 10 feet of stairs fallen. This is easily extrapolated for longer falls down stairs, but this module suggests that on this particular stair, a character is able to stop themselves about 10 feet from the point of falling.
Rise of Tiamat suggests 1d6 per 10 feet of stairs
In some caves in Rise of Tiamat, we see:
Steps are cut into the wall of the chute, creating steep, icy stairs that drop down 100 feet in a tight spiral. Because this entrance is seldom used, the steps become increasingly obscured by frost as the characters descend. At the 40-foot mark, a character must attempt a DC 12 Dexterity (Acrobatics) check to maintain a grip on the dangerously uneven footholds. Failure means the character loses his or her footing, sliding and tumbling 60 feet to the bottom and taking 21 (6d6) bludgeoning damage.
A 60 foot fall down these stairs deals 6d6 bludgeoning damage, so 1d6 per 10 feet is becoming a pretty consistent pattern here.