To ask whether one has the right to never be offended is a question that pretty much answers itself, at least for most reasonable folks. The fact that lawsuits are being filed, and won, on the basis of someone's manufactured offense over holiday decorations or a school mascot, is a sad indicator of rampant irrationality.
But beyond that, it occurs to me that taking offense is a sure sign of immaturity, something that the well-adjusted adult ought never to experience. Or if he does, it ought to signal something that he needs to work on. Not to advocate for the insensitive jerk who goes about deliberately insulting others, but to suggest that the mature adult ought not let such behavior rattle his cage, so to speak. Easier said than done, of course.
If I am insulted or ridiculed or criticized, it is either 1) for something over which I have no control (e.g. my race, height, age, etc.), or 2) for something I could change if I wanted to (e.g. my faith, values, political opinions, etc.) If it's for the former, as a mature adult, I ought already to be at peace with those factors outside of my control, in which case I can simply shrug off the insults or criticisms as pointless and moronic, and think no more of them.
If the criticism is for the latter, for something I can change, then there are two further possibilities: either 2a) the criticism is valid, or 2b) it is not. If not, I can again just shrug it off. If it is valid, then I ought to take it to heart. To be insulted in such a case is to say, in effect, that I prefer to cling to my inferior opinion and feel insulted, rather than adopt the better view.
And how to tell which criticisms are valid? Weigh the opposing views against one another, and use the faculty of reason to sort it out. Easier said than done. But it can be done. Therefore it must be done.