I remember some of the science/skeptic type of podcasts/blogs I listened to/read in the mid-00s often referencing the fact that vaccines are truly the victims of their own success. I think they are quite right from what I see and what is easily explainable human nature when it comes to risk assessment.
Sadly, I think it might take something like the current measles outbreak to disabuse these people of their lunacy. I'm Gen X, so I don't know that too much of this really took root with most of us, though we didn't really see much of these diseases first-hand either, but we did hear horror stories from older family members (I mentioned my grandmother elsewhere in this thread, but my mother was also a nurse and thought the antivaxxer phenomenon was sheer insanity, too). Though Jenny McCarthy is Gen X, sooooo....it seems at least some of our generation fell for it.
I think it might be the mid/younger Gen X/millennials that this kind of thing really began with. I doubt too many boomers or older X'ers really fell completely for the anti-vaxxer stuff, at least when raising their kids. I think not seeing first-hand, or at least hearing about, some of the terrors of these diseases from relatives you trust really does have an effect. Sometimes seeing something really does have an impact.
To compare it to something else - I remember older people and teachers telling us kids about 17 year cicadas. I think I didn't exactly disbelieve them, but it seemed really, really hard to imagine it to be the scale they claimed. It seemed like the older generations were exaggerating or maybe pulling your leg just a bit. Really? These things are everywhere, making all this noise? Until it happens.
There is little that can really prepare you for something like that, until you live through it. I think a lot of things in life might be like 17-year cicadas, including the fact that horrifying, awful diseases lie just beyond, waiting for the day when vaccination levels are not sufficiently high to keep herd immunity....and that younger generations sometimes have to actually listen to older generations.