If bad behaviors seen in public are not rebuked in public, then any private consequences won't be incorporated into the culture of that group of people (but the bad behaviors will become the culture).

Your response to bad behaviors should always be more visible than the bad behaviors themselves, and as quickly as you can reasonably react.

Start with immediately interrupting obvious bad behaviors: "That's not what we do here." Respond to more subtle issues by investigating the underlying factors, dynamics, and history to make sure you're addressing the right thing in the right way (and not just tone policing someone who's justifiably upset) — but bring clarity back to the original audience (or wider, to squash rumors from spreading) with a louder voice than any of the prior voices.

As Dr Strangelove says, "Of course the whole point is lost if you keep it a secret WHY DIDN'T YOU TELL THE WORLD, EH?"