This robot is small, purpose-built, and can easily climb stairs. The website has a lot of photos and is a great source of technical information for stair climbing robots. If you are thinking of building something similar, this is a must-read.
StairBOT is a small robot for indoor environments. On even floor it drives like many other small robots with a differential-drive. In addition it can change its length with linear guides mechanism with a spindle-drive. By this mechanism it can – together with its omniwheels (with brakes) and a support – reliably climb up and down regular sized stairs. It was one of the objectives for the design to use as few actuators and sensors as possible.
The StairBot – Link