I have a dilemma?

Also, can you put the cat in a crate---keep him away from the dog so that he doesn't get scared? Maybe try putting the cat in crate downstairs, and then the dog might want to go upstairs.

In general I don't advocate rehoming pets, but in this instance I think it might be in her best interest. It doesn't sound like an ideal situation for either of you. Greyhounds tend to be "couch potatoes" most of the time, but do need exercise and companionship. If you are able to, you could work with her to desensitize and habituate her to the stairs (use treats and praise to coax her up, step by step) and to your cat (as long as your cat isn't trying to hurt or scare her). You can also try interactive toys to keep her occupied, but it's probably not fair to her for you not to spend some time with her through the day.

Did you get the greyhound from a retired racing rescue? You should try to inquire with them, they may have some tips.

It sounds like the dog's unhappy. My first priority would be the happiness of my dog. In your difficult situation the two of you can't enjoy each other. If there's a rescue group or similar organization in your area why don't you consider contacting them & having a talk. They should be able to advise you what would be best for the both of you. You both deserve better that you have right now.

I would try desensitizing her to the stairs, like an above answer said, using really good treats that she likes. Also exercise (physical and mental) is always a good idea. Even for dogs that are considered lazy, if their brain is not mentally stimulated, they can act out in destructive ways; 15 minutes basic obedience would be a good start to mentally stimulate/challenge your dog and will build trust and obedience possibly making the stairs an easier task to tackle. Exercise, hiking, meeting other dogs at the dog park always gives our dogs enough stimulation to curb some of the destructive behaviors/boredom. If you feel you don't have the time or patience to work with her, please don't give her to just any shelter. Some shelters euthanize even though they say otherwise. Please try to find a rescue, maybe even breed specific, that will be able to find her a home.