Of course the BOF answer is to walk down stairs - if they had wanted you to ride down them, they would have put a ramp there.
More seriously, If I have understood the question correctly, you are hopping or dropping down from step to step on a flight of stairs, landing perhaps a little heavily, and your heels are being pushed lower than the balls of your feet, thus pushing your foot upwards towards your shin. Am I right?
Assuming so, I can answer this two ways: as a unicyclist, I’d say whenever there is danger of the pedal kicking your foot back (I get it on sudden steep hills) pedal with your instep. This will reduce the leverage exerted on your foot.
But as a dancer, I’d say the problem is one of how you are using your legs ankles and feet. Your legs are complex mechanisms, with flexibility at the hips, knees, ankles, and the balls of the feet. Of these flexible points, the knees, ankles AND balls of the feet are all useful for spreading the load when you land suddenly.
Forget the uni for a moment, and imagine jumping off a chair onto the floor. You will allow your knees to bend slightly to absorb the impact - otherwise, you’d jar your spine.
Now imagine your jumping onto the floor and you don’t want to make a noise. You’ll probably try to land on ‘tip toe’ and stop your heels hitting the floor. You’ll let your feet roll slightly to absorb part of the impact, and your knees bend to absorb part of the impact.
In an ideal landing (e.g. when doing the Morris dance step known as a plain caper ) your leg will act like a three part ‘concertina’ shock absorber, with the three ‘hinges’ (ball of foot, ankle and knee) all working together to share the sudden load.
This requires practice and technique, and it requires anticipation. You need to be ready before you hit the floor. It’s important for the various muscles to be ready but RELAXED - if you are tense, you will hit the ground with ‘wooden legs’ and hurt yourself.
Think also of parachutists, who bend at the knees and roll with the fall - both things happening in one smooth motion. Think of crumple zones on a modern car. The car doesn’t stop dead on impact, but crumples, dispersing the force in a planned and controled way.
So apply this to your jumping down stairs problem: I’d suggest practising on small single steps and get the feel of your ankles, feet, and knees working together to absorb the blow in a controlled manner. Imagine you’re trying to land quietly on a wooden floor, or that the floor is only just thick enough to bear your weight. When you get the feel for it, you can build up to the bigger drops.
Of course, I could be wrong, but I guess from your description of the symptoms that you’re landing with your knees tense and most or all of your momentum is being transmitted through your ankles - and that’s a lot to ask of an ankle.