Looking on the internet deeply has found these results:

door to the stairs is the most popular phrase on the web.

door for the stairs 15 results on the web	door to the stairs 16,980,000 results on the web More popular!
Some examples and use cases from the internet:	Some examples and use cases from the internet:
So you have to go out the door, down the stairs, past the econ building, through the quad... You'll find the key to this door under the stair carpet here. Through the double doors, up the stairs, first door on the right. There's a storm door by the stairs. But every time I open the door looking for the stairs, all I find is a teeny tiny bathroom. Near the fire door, up the stairs. We take it from the stairs, doors, wherever we see a piece of iron. I need to go out that door and up the stairs	Why is there a door to this stairs? We take it from the stairs, doors, wherever we see a piece of iron. Near the fire door, up the stairs. There's a storm door by the stairs. Through the double doors, up the stairs, first door on the right. He told me to enter the house, cross the courtyard... and look for a metal door to the left of the stairs. You'll find the key to this door under the stair carpet here. But every time I open the door looking for the stairs, all I find is a teeny tiny bathroom.

door for the stairs

15 results on the web

door to the stairs

16,980,000 results on the web

More popular!

So you have to go out the door, down the stairs, past the econ building, through the quad...
You'll find the key to this door under the stair carpet here.
Through the double doors, up the stairs, first door on the right.
There's a storm door by the stairs.
But every time I open the door looking for the stairs, all I find is a teeny tiny bathroom.
Near the fire door, up the stairs.
We take it from the stairs, doors, wherever we see a piece of iron.
I need to go out that door and up the stairs

Why is there a door to this stairs?
We take it from the stairs, doors, wherever we see a piece of iron.
Near the fire door, up the stairs.
There's a storm door by the stairs.
Through the double doors, up the stairs, first door on the right.
He told me to enter the house, cross the courtyard... and look for a metal door to the left of the stairs.
You'll find the key to this door under the stair carpet here.
But every time I open the door looking for the stairs, all I find is a teeny tiny bathroom.