How is it even possible to be an antivaxxer in the 21st Century? This is a question you will not be able to answer simply, or even in a way that consistently covers all of the reasons for all of the members of this misinformed group. Instead, take the steps outlined in the articles below, and you might be able to talk some sense into people that appear to believe in nonsense.
The first step is to educate yourself. The first article outlines some useful facts and statistics about vaccines, their humankind-improving benefits and the very low risks associated with them.
Next, try to consider the issue from the perspective of a person who is paranoid that there is some big conspiracy against them, or that is so concerned with their child’s health to the point that they’ll follow any advice that they read on the internet without any serious consideration or deliberation.
Of course, that is a difficult position to be in, and it is perhaps even harder to imagine how a rational, reasonable person might end up like that, but then again, look at how many millions of people believe in things that don’t exist and have never been proven to exist!
After that, you should choose a trust-building strategy that allows you to broach the subject where you are both able to keep control of your own emotions.
The second article linked to below says it well when they state that you need to keep the dialogue open. If you accuse someone of being an idiot, or continually tell them that they are wrong, it is just as likely that they will double down on their beliefs, as per what we saw with the article on confirmation bias. You need to help people see the light of reason through the fog of superstition and miseducation by inspiring them to question their own beliefs, and realising for themselves which of their notions are mistaken.
Lastly, think big picture, not just about anti-vaxxers, but about all of the irrational, misinformed and harmful beliefs that are followed, supported or even tolerated in your community, and consider how to educate others so that they do not have to continue living their lives in constant a state of ignorance, paranoia and misunderstanding.