Sitting your trot without stirrups should be working mostly the same muscles as posting with them. All of the muscles you use when posting are used when sitting (including your thighs and your, well, bum), but you also use your abs when you sit, and I find that essentially everything is amplified when you drop your stirrups. This is going to improve your balance and your seat. You should NOT be gripping with your knees; it's easy to start clutching with your knees, but this will only make it harder to keep a good deep seat--it will make everything seem bouncier. Instead, think of tightening the very highest part of your inner thigh and grip with your bum. You shouldn't tighten your lower leg (your calf) at all unless you are asking your horse to move out or pick up the pace. If you start gripping with your lower leg for stability, you will be sending mixed signals to your horse, and he/she won't know what you're asking for. You should be able to pull your lower leg up behind you and hold your legs at your ankles and keep a steady seat. Don't try that unless you are with a trainer, though! That exercise is only done if your horse is on a lounge line.
My trainer used to do exercises with us to work on developing a strong seat, which is why you practice riding without stirrups, and I still use them from time to time. Without stirrups, ride in a slow to medium paced sit trot. Hold the reins in one hand, and with the other hand reach back, twisting at the waist, and touch your horse's hip. Keep your hand there and feel the gait, sitting deep in your seat. Then switch hands, so you are even. There's another exercise, which you should only do when supervised by a trainer or on a lounge line. Point both arms straight up to the sky, with no stirrups. If you are pointing straight up, you will be forced to drive your seat deep in the saddle, helping you to really feel the gait and get a good sense of what a good seat feels like. Then when you bring your arms back down to a normal position, you can try for that feeling.
Riding bareback has really been the thing that made my seat more stable, but you shouldn't just jump on bareback immediately. You need to make sure you are ready for it, and the horse you ride might not have ever been ridden without a saddle, so you need to talk to your trainer about it first.
At home, do lunges and squats, and something I don't really know what the name for it is, but I call "toe raises." For these, stand on a step with the ball of your foot on the edge, as if you're standing in the stirrups, and allow your heels to stretch down. Then point your toes, raising yourself up, but only at the ankles. This works your calves, which you shouldn't be working when riding without stirrups for balance, but certainly helps for using leg on less-than-enthusiastic mounts. Remember to stretch, though! Crunches can't hurt, either. Like I said, you do work your abs when sit trotting.
10 years riding