Cats need to scratch stuff but it's not to sharpen their claws. Their claws grow and they need to get rid of the outer layer - you've probably seen these lying about. There is also a territorial issue and they're marking their spot. It may be a lovely new home to you, but to a cat it's literally unknown territory.
First of all how does you cat scratch? Do they scratch horizontally (ie. the flat of the stair in front of them) or do they scratch vertically (ie. the side or edge of the stair above them)? To a cat, a carpeted stair is a great activity centre.
Cats have different scratching preferences so you need to provide a suitable alternative. Some cats like vertical scratching posts, some like ripping apart sisal floor mats. A common problem with most scratching posts is that they're too small for adult cats. There is a jumbo scratching post available called the 'ANCOL Fat Boy'.
Some cats don't like the sisal string used on cat scratching posts. You could try using the felt underlay on the underside of carpets. Rub in some catnip to make it interesting.
Lastly, you could try the hierarchy thing. Praise them for using the right place and shout at them for using the wrong place. They won't understand but will rationalise it in terms of territory - yours and theirs.
Good luck.