There is some evidence (sorry, no cite) that when people are given evidence refuting their beliefs, it only hardens their position. As noted above, you cannot reason someone out of a position they didn’t reason themselves into.
I’ve also heard claims from religious types that we are opposing god’s will by a) using vaccines or b) fighting climate change.
People with ridiculous, unsupported beliefs have always existed.
The internet has done two things: it has let these people find each other, and algorithms created by the advertising companies* these people rely on to reach one another encourages them to fall ever-further down the rabbit hole.
*Google, Facebook, and their subsidiaries.
Stupid people who do not realize they are stupid. They actually think they are smarter than your average bear. These are folks who make a deeply flawed argument and never even consider that they could be mistaken.
Some folks make it a proud announcement of ignorance. “I know this and nothing will convince me otherwise”. They may as well say “I will hold on to my stupidity and will not be swayed by facts!”
My Grandma used to have a refrigerator magnet that read:
God said it.
I believe it.
That ends it.
She was totally perplexed when I told her that the first three words were effectively unnecessary.
There do exist people who hold these ideas purely because they lack information and can be convinced. I’m not sure what the relative percentage is although I suspect that it is smaller on the internet than it is in real life. For such people A quick citation may be able to debunk what the misinformation that they have heard and set them on the right track.
So if someone writes “I don’t vaccinate my child because I’ve heard it causes Autism and I think better safe than sorry.”
It is probably worthwhile posting a link to the CDC that strongly advocates vaccine and maybe one that debunks the Wakefield study. If they respond with some reasonable easily debunked concerns, for example that the diseases that people vaccinate for are so rare that even without the autism connection its still not worth it do to the side effect, a second post may be warranted to address those, but if they start talking about the evils of Big Pharma hurting children for profits, or if they keep bringing up different reasons to stick with their position, give up you aren’t going to change their mind and are just going to make them feel persecuted.
Still I think an initial debunking post is worthwhile. Even if you don’t convince the person you are arguing with there might be others who are less firmly planted in their beliefs who might believe the original poster if you leave their ideas unchallenged.
The problem is that there is conflicting information, and people are not good at deciding which is valid. As far as I’ve seen, knowing someone whose child got autism supposedly after being vaccinated - or even seeing someone say this on YouTube - trumps any amount of statistics from the CDC or WHO.
Instead of showing that, maybe talking about children who died from the measles would be more effective. 110,000 children died from the measles worldwide in 2017, I bet most anti-vaxxers don’t know that.
An anecdote carries a lot more punch to it. It personalizes things. “My son got autism after the vaccination” rings with greater resonance than “a hundred thousand children died of measles last year.” The latter, presented mathematically, comes across as a statistic, which has little emotional effect.
The solution is to present preventable, vaccinate-able, diseases in graphic nature. Bring out the anecdotes of “so and so died an easy, preventable death that could have been averted by a vaccine”.
That would not impress the typical antivaxer, who would respond:
*good grammar and critical thinking skills are inimical to the antivaxer set.
And I can’t properly number lists, but hell, it’s late and I’m beat.
Nah, it just adds authenticity. Though next time consider more caps. 
This one really gets to me. My wife wrote two pro-vaccine books without one penny of funding from big pharma.
Damn it.
I’ll still fight ignorance diplomatically but persistently, but I’m focusing on minimizing the dangers anti-vaxxers impose on the public. Washington state is considering removing its ill-advised philosophical exemptions AND the standard religious exemption, so I contact state legislators often to try to counter the efforts of the anti-vaxxers, who not only rallied on the Capital steps not long ago but penned an open letter calling vaccinations “medical rape” and adopting the #ustoo hashtag.
There are loopholes, of course, and anti-vaxxers will use them. In WA, naturopaths, who are vehemently anti-vaxx, are allowed to be primary care providers. They’ll happily falsify documents to get medical exemptions.
Think how much we could accomplish if we weren’t fighting so much ignorance so much of the time. It’s exhausting. I don’t know how Uncle Cecil did it.
Short and to the point. Perfect.