The carpet on my stairs is worn. Should I repair it or just remove it?
I'd go ahead and pull it up and see what you've got.  It sounds like it's in pretty poor shape so you're going to have to get rid of it either way.

Depending on how the carpet was installed you'll probably find tack strips that look something like like this: 


You might also find staples -- I don't know if this is common but our installer used a staple gun in some troublesome spots.  Most if not all of the staples will probably be pulled out when you rip out the carpet.

Before removing anything, take a look at the hardwood stair treads and make a preliminary decision about whether you think they're in good enough condition to leave exposed.  If they're really rough and/or you want to recarpet, leave the tack strips in place and hammer in the staples.

If you do want to leave the stairs uncarpeted, you can just remove the tack strips, being careful not to damage the floor underneath.  For minimal damage I'd recommend this type of prybar available at box stores for ~$14:



Use a hammer to tap the pointy claw end (like you would hammer a chisel) under the head of each nail holding the tack strip down, then pry up the nail enough that you can get your hammer's claw under it to pull it out.  No tools ever need touch your stair tread.