It is well-established in fiction — as in real life — that falling down a flight of stairs is dangerous. A significant tumble can seriously injure or kill a person. "Pushed down a flight of stairs" is by now a hackneyed modus operandi in crime dramas. Recognizing that D&D is not a physics simulation, how should a DM model that danger?
Under normal falling rules, "[a] fall from a great height" deals 1d6 damage per 10 feet fallen. (PHB p. 183.) That wording seems to presume a freefall ending in a single, hard impact. Falling down stairs is at least arguably different: it doesn't involve the same velocity, but it might involve a lot more bludgeoning.
Do the normal falling rules cover falling down stairs? Are there any alternative rules in published 5E materials? If it helps, our table does use the optional falling rules from Xanathar's Guide to Everything, although so far we haven't treated those rules as answering this question.
Best Answer
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:
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:
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:
A 60 foot fall down these stairs deals 6d6 bludgeoning damage, so 1d6 per 10 feet is becoming a pretty consistent pattern here.