I think she is adressing something more than good sense or reasonable thinking: she is describing what it means to say "this should not have happened" and "you should not have done that". Judgement, for Arendt is qualifying an event as unacceptable and those who participated as responsible. She is also confronting those who witness said event and stay silent and are unwilling to pass judgement on those who committed the deed. Thus, she is passing judgement on those unwilling to judge.
Judgement, for Arendt, is fundamentally a moral act that defines our moral responsibility; and by judging, I think she is saying, we define what it means to be human. And, being human (among other things), means having to judge the deeds of ourselves and others...and denying this process is to deny our humanity.
At least that's what I'm getting so far.