It's very common to cover stairs in carpets and rugs. Sometimes they are covered in wall to wall carpeting, sometimes with a stair runner which is a long thin carpet that goes down the center yet leaves the edges of the stair treads exposed, and sometimes with small individual rugs per step sometimes called tread rugs. Why is covering stairs with these so popular? For example every staircase in my home as well as my office building is covered by carpets. There are several reasons that we'll examine below.
1) One of the top reasons in certainly for looks. There are thousands of different colors, patterns, and materials available for options and the looks can be quite dramatic. Especially with many stairs being quite prominent in many homes, a lot of money as well as time and effort are often spent finding the perfect rug or carpet for the master staircase. In particular, both stair runners and rugs leave much of the staircase exposed which is great when they are made of a beautiful material like an exotic hardwood, stone, or marble. It would be fair to say that decor is the top reason for most people in residential environments.
2) Another reason is safety. Rugs and carpets can have non slip capabilities, certainly more so than hardwood for example. Many people fall or slip and are hurt on stairs every year so this can be significant. Of course there are also plenty of other non slip materials that can be used as well and other things like handrails and proper lighting can make them safer as well. Safety is a major factor with stairs, both inside and out, and there are stair carpets and rugs appropriate for outdoor use as well.
3) Rug and carpet has sound deadening qualities, and people (or pets) going up and down stairs tend to be very loud below. This is sometimes overlooked but can be very significant. One friend added a carpet runner specifically so that their dog wouldn't make too much noise using the stairs at night. Also, children tend to run on stairs making lots of noise that can be somewhat limited with a good carpet and carpet mat underneath. The amount of sound absorbed can sometimes be surprising.
4) Construction is expensive. Construction grade material is less expensive than finish grade, and often instead of using finish grade materials such as hardwoods on commercial real estate, the unfinished stairs are covered by a commercial wall to wall carpet. This not only can cost significantly less initially, but the carpet is also less expensive to replace than worn stair treads.
5) Stairs are heavy traffic areas. Heavy traffic go hand in hand with ordinary wear and tear as well as chips, dings, and bruises. Any covering, whether rubber stair treads or a more elegant carpet or rug can help protect the stairs.
Stair treads and carpets are extremely popular for the reasons listed above as well as other less common concerns. Often it is chosen for one reason, but will fulfill several of the above.