When a cool character Homura came across a highly volatile situation, she left a wisely sayingid "I'm always on a calm people's side, and an enemy to foolish people who waste [their] time on meaningless battles.

This saying deserves to be delivered toshould be heard by peoples who can't removovercome their anger and accept different opinions.

"Delivered" is not the best verb for use with "a saying".

"Peoples" is a valid word, but refers to nations ("Peoples of the world"). I think "people" (individuals) works better here.

I also want to become such a calm person and be wise enough wiseto judge statements by myself so that I won't be misled by rumors and judge statements by myself.

I moved the "judge statements by myself" part of the sentence ahead of the negative "won't" so that the sentence can't be read as "... that I won't be misled by rumors and I won't judge statements by myself."

All we have to do is to behave with our cool judgement.

When a cool character Homura came across a highly volatile situation, she left a wisely sayingid, "I'm always on a calm people's sidethe side of calmness, and I'm an enemy tof foolish people who waste the time on meaningless battles.

Which side are you on?

This is a bit ambiguous.

I also want to become such a calm person and be wise enough wisesuch that I won't be misled by rumors and to judge statements byfor myself.

Needless to say, to be a calm person doesn't mean to bhat you are indifferent to political issues.

I think the original makes grammatical sense, but it's a bit awkward.
It's okay to use second-person pronoun In English though (as opposed to Japanese).

All we have to do is to behave with our coolcool down before making our judgements.

'to behave with our judgement' is a strange expression. I've changed it to what I find natural.