I've frequently seen that the difference between exacerbating and at least not-further-harming knees can come down to proper stances and footwork. For example, I emphasize not pushing the knee past the ankle during a lunge as a way to help protect the knee.
What are good drills or good things to emphasize during stances and footwork to help avoid exacerbating non-specific knee problems?
Knee instability is usually as a result of one of two things - You either have a pre-existing injury that is contributing to the instability, or you have muscle weakness that is contributing. (This is assuming no congenital defects).
For the first, you may have to supplement with braces and/or corrective surgery. While you can protect the area with proper form, even a correct stance may make a pre-existing condition worse.
For the second (And to some extent the first), the stronger you can make the muscles/tendons that pass through/around the knee, the more external stabilization you will have. This may require supplemental weight training, which I suggest you do under the guidance of a certified (NSCA, CSCS, etc) trainer.
As far as the training, you can do pretty much any of the various agility drills with ladders, cones, etc. with the caveat that they will increase the chance of injury. Proceed slowly with great emphasis on correct form, so that as you eventually get faster your form stays correct. The pivot drills will be the most risky, as you are rotating around a fixed point, lateral movements would be the next step down on the risk factor scale, while forward/back linear movements are likely to be the least threatening to stability.
However, ANY training that is done at high or competition speeds carries its own risks, as even a slight misstep may cause a pre-existing condition to flare up or get worse. In the end, you need to really be aware of your own body and judge what might or might not help/hinder your progress.
Taekwondo, especially, seems to be a great destroyer of knees. Pretty much everyone I know at high levels in the art has experienced knee injury at some point. I think this is because of the right angles the stances and footwork place your feet at. Because of the way that TKD movements are structured, placing weight on the heel of the foot during a pivot will put "drag" on the knee, causing it to twist painfully. It is absolutely essential to pivot only on the ball of your foot when doing pivots for kicks.
One thing that can help with this is to do lots of heel raises, focusing on balancing on the ball of the foot. Also, doing the footwork without setting the heel down will contribute to emphasizing the ball as the pivot point for stepping and kicking.
I don't know if this answers your question generally, since I don't know what the requirements for weight placement (i.e. heel/ball) are for the art that you do. It's just been something that I've had to deal with in training TKD. Hope it helps.
This answer is mostly just in support of the others...
Muscle strength is the best way to overcome a relatively minor knee problem and avoid future injuries. I partially tore my ACL about 10 years ago and had constant knee pain until I started spending 3 to 4 hours a week practicing Tang Soo Do.
Four specific exercises that greatly strengthen legs and thereby stabilize knees:
"Sissy squats"- quick repetitions of squatting and rising (one of my instructors is a former strongman competitor, I imagine any exercise not involving 400 lb weights has "sissy" in the name somewhere)
"Sissy kicks"- rapidly squat and rise, keeping 1 knee bent as you rise and execute front snap kick, alternate legs
Squat and hold- try to work up to 5 minute holds, there are several subtle ways to vary position to alter muscle group effects
Last one is harder to describe:
Spread your feet shoulder width and hold out your arms to the sides with forearms pointed upward. On the balls of your feet pivot 90 deg left or right and lower the back knee until it almost touches the floor. Rise up and pivot on the balls of your feet 180 degrees and repeat. 30 seconds of these can be grueling depending on fitness level.
Take this one nice and slow until your knee stability improves.
Others have given good advice on preventing injury whilst practicing, so I'll tell you what to do when you're not: get yourself a stationary bike and use it for at least half an hour per day on medium intensity to rehabilitate and strengthen the knee ligaments. Or if you're like me, get a real bike with clip-in pedals and go for a nice ride with gentle inclines. It's also great rehab for bad ankles and leg muscles.
