Though many carpet cleaners and shampooers are manufactured, they don't all work well on stairs. You want to have a steam cleaner that truly gets down into the carpet and pulls out the dirt. This means you want a cleaner that heats the water, not one that you just adds hot water to the carpet. Not all steam cleaners are built for that, or come with attachments for use on stairs.
Rug Doctor machines are available for rent at many retail location such as grocery stores or home improvement centres. The machines are steam cleaners that pump steamed water into the carpet and pull it back out along with the dirt and grime. They have a hose attachment with a hand tool, so that you can work on the stairs one at a time and get into the corners.
Hoover SteamVac with Clean Surge
The Hoover SteamVac with Clean Surge is recommended as a fairly capable steam cleaner for carpets. It has a hose attachment that will get to the stairs, and it is not as leaky as some other steam cleaners in the Hoover line. The main complaint about the cleaner is the plastic parts, which may wear out faster. However, the cleaner itself is fairly reliable and easy to use.
If you prefer not to deal with a cumbersome steam cleaner on the stairs, you might consider a hand-held version. Hoses on the full size cleaners, don't normally reach from one end of the stairs to the other, which means at some point, you will probably be balancing the cleaner on the stairs while trying to clean. A hand-held steam cleaner is easier to use on the steps and works well on stains. Consumers tend to lean toward the Bissell SpotBot Deep Cleaner. It cleans with the bottom scrubber as well as a hose piece that will get in the corners.