Sometimes my characters say things I disagree with and do things I never would. Rarely, I censor them, for example if it wouldn't be that big of a deal, wouldn't hurt the story or take away from their character. But usually I let them be who they are This comes with some risk.
If a protagonist says something I disagree with, I worry that my readers may think I would also say something like that. Oddly, I'm more worried about offending people I do know as opposed to complete strangers.
Sometimes my writing is an outlet for things I believe in. My characters can't help but reflect my values. And if my values offend, so be it. But when, in the act of being themselves, they say something I completely disagree with, that I think may offend others, it's a little scary.