It may not seem important to a beginner, but the more you think about it, the more it makes sense.
After all, every step and position in ballet has a specific technique, why would holding the barre be any different?
Place Your Hand on Top
Let’s go back a little. Rather than thinking of “holding” the barre, think of naturally placing your hand on top of it as tension free as possible. Also, you want to actively think of not “using” or relying on the barre to accomplish any steps or positions. Just think of it as an aside or accessory to the natural progression of a ballet class, not a necessity to do a step.
There is no tension, just simply placed on top. Also, you will notice the thumb should be placed close to the fingers. The idea is that if you lifted your hand off the barre, it would already practically be in a correct balletic shape, not looking like you are holding a soda can. Your thumb doesn’t need to be touching your fingers, but it also shouldn’t be on the bottom of the barre.”