The phrase "back pedalling" can cover a number of techniques.

I don't skip or skid (intentionally) because I ride with expensive tires, and I think I can stop faster by back pedalling than by skidding.

Back pedalling works best with gear ratios below 72 gear inches.

And, back pedalling involves more technique than it does strength.

Really, nothing about a fixed gear bike should cause knee injuries.

If one feels symptoms, then the fixed gear bike has only revealed a pre-existing problem, and has not caused the problem.

Riding a fixed gear bike, regardless of technique or style, involves much lower stresses and loads than the knee experiences while running or going up and down stairs.

In any event, one should have a saddle position low enough to allow the full drop of the heel with the pedal at the bottom of the cycle.

This applies on any bike.

Most people ride with their saddle too high.

Look at professional road racers.

They sit much lower than the average non-elite rider.

When I back pedal, I drop my heel as much as possible, and I allow the aft rising-pedal to rise at a controllable rate.

I also keep my seat barely in contact with the saddle.

If I try to stop the pedal, I can't, and I actually become less efficient.

I don't (at least not very often) try to keep the front descending-pedal from descending.

Rather, I concentrate on dropping my heel, keeping it dropped, and controlling the rise of the pedal.

If I put any upward force on the front descending pedal I will often times skip or skid, and, besides wearing out my tires, a skidding tire does not stop as fast as an adhering connected-to-the-road tire.

Above 72 gear inches, either learn to skid (and wear out tires), or use a front brake, or both.

Remember, drop the heel and keep it dropped, and keep your bottom lightly in contact with the saddle.

Let the pedal rise.

With practice, you will find that you can more effectively slow down the bike with good technique than you can with brute force.