Would you be able to put up a gate by the stairs?
Our Bodhi has high prey drive, and if our cat runs, she simply sees it as prey. If the cat sits still, she'll leave it.
What steps have you taken to introduce the two, and to tell your dog that the cat is part of the pack and off-limits?
What steps have you taken to break the dog's focus on the stairs or to redirect the dog's behaviour?
(and of course, pictures of your cat and dog would be much appreciated!