HazardsMind is right. An Arduino is overkill. But with an Arduino, you have a chance to do something more than just lighting up the stair that is being stepped on. How about something like this.
As you step on the first stair on the way up or down, use PWM to ramp up the brightness on the LED strip for the first stair to full.
At the same time, ramp up the brightness on the next stair to say, 2/3 brightness, and the second one after that, up to 1/3 brightness.
Then for each stair you vacate, you ramp that one down to 2/3,, and the previous one to that,down to 1/3.
After the 1/3 behind the person, ramp down to OFF.
The result should be far more interesting than just snapping them on/off. It should make a nice 'moving wave' effect centred on the one climbing or descending.
It would be a bit of a challenge though, and you'd have to decide what to do with 2 or more folks on the stairs at the same time.