Being offended is a sign of weakness.
When someone does or says something and you feel offended, it's a sign of weakness in yourself that you should investigate thoroughly and resolve. Let's talk through a scenario to better understand this idea.
Let's say that someone uses a racial slur towards one of my kids. Anything I say in response to this person is going to be about as effective as a flame war on the internet so there's really no benefit to it besides maybe briefly feeling better about myself. Now, if I actually engage physically based my feeling of offense, chances are that I'll wind up in hand cuffs or in the hospital.
Allowing myself to react based on that perceived offense means that I'm letting the other person control me. They get to decide how I'm going to think about myself. They get to dictate my behavior. That's weakness.
But what about my kid? What about how they'll think or feel about themselves hearing someone speak to or about them this way? Shouldn't I do something to protect them?
Yes. Of course. In fact, that's exactly what I'm doing. By rejecting the person's attempt at offending me, I'm showing them how to be strong enough to control anger and reject the control that other people try to exert over me. I'm setting the example I want them to follow.
It's reasonable to assert that such a complex and potentially subtle lesson may be missed by some children. If you feel like your kids might instead see you as being weak and unable to stop the mean person, take the time to explain what you're doing and why you're making the choices that you're making. It's even more powerful if you do it in the moment and in front of the person that is causing the offense.
"A lion doesn't concern himself with the opinion of the sheep." - Tywin, GoT