we can say he ran down the stairs but we cannot say he ran down stairs.On the otherhand we can say he ran downstairs .
Ran down the stairs means, the DIRECTIONof running and ran downstairs means, the direction of the place of the stairs in that, he did not run the upstairs one. I think each has its own purpose.
both are same
He ran down stairs, up ladders, over bridges, anywhere he could! He ran downstairs to answer the phone. Both can be used, depending on context and the writer's intent.
John ran down stairs to get downstairs. So he ran downstairs by running down stairs.
These sentences seem grammatical to me.
Either could be used, it;s more comfortable...