In the case of anti-vaxxers (most of whom are probably sincere,) when you become upset or even infuriated with them it cements their beliefs. They interpret your anger as evidence of their position being right. Therefore if you want to change their minds, the first thing you must master is your own emotions. Hiding your anger isn't enough, because people always leak information about their emotional state. You need to find a new frame of mind that allows you to have discussions with people who are wrong without becoming upset. I know this is difficult with antivaxxers since their beliefs put children in harms way, but it's necessary.