Yep, you will always take more steps when walking, as long as you are moving with the escalator.
If you run, a higher proportion of the distance travelled will be by you taking steps. If you walk, a higher proportion of the distance will be made by the moving escalator.
Say there are 10 steps, and it takes 10 seconds for each step to go from the bottom to the top (1 step/second). If you walk at a speed of 1 step per second, you'll reach the top in 5 seconds, because you are moving at 2 steps per second (walking speed + escalator speed).
If you run at 4 steps per second, you'll be moving at 5 steps/second, due to the movement of the steps. This means it'll take you 2 seconds to reach the top, which means you will have actually made 8 steps.
8 > 5, therefore, you take more steps by running than by walking.
The faster you move, the more steps you take.
Steps taken = distance (steps) / walking speed (steps/second) + escalator speed (steps/second)