It becomes heat, in your muscles, and that is the whole answer — but the mechanism is more interesting than the destination.
On each step down, you fall a short distance and gravity does work on you, which would speed you up. You do not speed up, so something removed that energy. That something is your leg muscles contracting while being lengthened — your quadriceps are pulling to hold you up at the same time as the knee is bending. This is called an eccentric contraction, and it is your body acting as a brake.
A brake turns motion into heat. A car's discs get hot; your thigh muscles do the same thing on a smaller scale. Some of the energy also goes into deforming tendons and cartilage and into sound.
So the energy is not vanishing and it is not being stored. It is being deliberately destroyed, by you, on every step, to stop you accelerating down the stairs.