Good badminton body balance is very important AT ALL TIMES when you're playing in games. It's important for:
Better Footwork: After a shot, you must have good balance to quickly return to your base and prepare for the next shot.
The KEY to badminton body balance is to make use of your non-racket arm.
When you perform any badminton strokes and shots, use your non-racket arm to balance the weight of your racket arm, as well as the force of your swing motion.
I'll show you an example on how to use your non-racket arm for better badminton body balance while doing a badminton overhead stroke.

The picture above shows you an overhead stroke. While you're positioning your racket arm to hit the overhead stroke, position your non-racket arm to the position as seen in the picture above.
DO NOT swing your non-racket arm forward or else you'll lose balance. Slowly extend it forward. This is to counter the weight of your racket arm.

Just when about you hit the shuttle, slowly stretch out your non-racket arm until it's almost straightened (see picture above). This is to counter the force of your swing.
As you swing your racket forward, slowly return your NON-racket arm to the side of your body.

When you stretch out your non-racket arm for better balance, gently point the wrist of your non-racket arm upwards as shown in the picture above.
Try it now! Bend your wrist upwards and move your arm around now, you'll realise it's a lot easier to control the movement of your arm compared to not bending your wrist upwards.
Avoid tightening the muscles of your non-racket arm. Just allow your wrist to naturally bend upwards.

The badminton jump smash will require you to maintain extremely good body balance since the stroke is performed in mid-air.
The picture above shows the non-racket arm stretched out for better balance.

In the picture above, the player lunges forward to perform an underarm stroke. If you lunge forward like that, you'll most likely lose balance.
Therefore, stretch your non-racket arm towards the opposite direction of your racket arm to create more badminton body balance.
Extend your non-racket arm to the opposite direction of your racket arm when performing any badminton strokes.
However, extend it gently, don't swing your non-racket arm or else you'll lose balance.
As your swing your racket arm, move your non-racket arm back to the side of your body.
Do not tense your non-racket arm! However, bend the wrist of your non-racket arm slightly upwards (as seen in the pictures above) to have more control over the movement of your non-racket arm.
Make use of your non-racket arm wisely. You'll improve your balance when you start to know how to make use of your non-racket arm when executing shots.