I guess that's a vague question, and I guess the answer depends on the situation.
If someone asked me this question in general, I'm inclined to think that the high road is usually choosing to ignore something offensive rather than doing or saying anything in response. The idea is that you don't "stoop to his/her level," assuming there is a "his/her" in this general situation. You take the high road by diffusing the situation instead of fueling it.
In general, that sounds pretty good.
In general, I struggle with this. Maybe not in the immediate situation, but definitely after the fact. If someone says or does something to offend me - someone I don't know well - it may appear that I take the high road because I don't always know how to react. But inside I'm fuming, and a few minutes too late I think of a few perfect things to say in response to whatever was said or done. So perfect that I really want to walk back up to the person and lay it on 'em. It'll be clever and smart with a little zing, and the person will feel bad about whatever he or she said in the first place.
What? No.
If you can respond with that clever, smart little zing without skipping a beat, maybe you'll have some impact. Or maybe (probably) you'll just spark an argument. But coming back with a delayed response to the offender? Someone you don't even have a real relationship with in the first place? There's nothing to gain, and the "perfect" words you've been stewing over probably feel foolish as soon as they leave your mouth.
So, as far as this person knows, I said nothing, and I'm not writing a situationally-inspired blog post about them.
...but for the record I think this person sucks, and I think the words I didn't respond with are awesome.
XOXO,
Kristin