There are three ways to post, only one of which is correct.
The first is by using the stirrups to hoist oneself out of the saddle. To do this, the rider presses his foot against the stirrup and straightens his knee. The most obvious sign that the rider is doing this is that his lower leg will swing forward as he rises out of the saddle. This is usually the first way a beginner will attempt to post. An unfit rider will also default to this. Many riders do this, just not as obviously.
The correction lies in taking the weight out of the stirrups and firming up the thigh to support oneself. This is why posting without stirrups was once so commonplace as a correction.
The second way to post is by lifting oneself up by the midline. Imagine a hook pulling up from under the sternum (where the ribs join in front). This results in an elongated front, a hollow back and an unstable rider. This seems to be more common in women then men (who are more likely to press on the stirrups).
To check this out, I have the rider place a forefinger on the hollow below the sternum and see if this gets tight as they rise. It should stay soft. This is a much harder pattern to overcome. The rider may believe this is the correct method of rising. The use of these muscles is usually ingrained.
The correction lies in keeping the chest down while rising. This often contradicts what other instructors have offered. The new way of posting may feel slouched and just plain wrong. Along with keeping the chest down, I focus on keeping the lower back straight (it will feel pushed back to the rider). Again, the thighs need to contract more.
The correct way to post is by contracting the buttocks and thighs.
The current practice of teaching riders to relax their legs results in a lot of riders being very unstable. In dressage, the thighs are turned in and are the entire thigh from hip to knee is firmly on the saddle. It is the outer thigh muscles that are mobilized (turn your heel away from the horse). In jumping, the rider’s toe is turned out from the horse about 35 degrees (with the baby toe lifted) and as much of the rider’s leg is wrapped around the horse from the thigh to the ankle.
Many female riders seem to never use these strong muscles at the back of the thigh. It is useful to put the reins in one hand and to grasp the back of the thigh right where it meets the buttocks. At a walk, practice contracting and relaxing the muscle (grasping it makes it easier to feel when you have it right). Some riders need to do this even in the trot.
Once a rider can reliably use the correct muscles to post from, other things can be worked on. The knee must point down to the ground, not forward. The thigh should stretch out of the hip socket. Finally, the toes spread while rising, resulting in a lowered heel.