Regarding the first part of your question, I think that the single greatest factor that led to an anti-vaccine backlash (which began in the early 1800s in response to the smallpox vaccine) was vaccine mandates. I think if you ask the professional anti-vaccine groups what would cause them to stop their activities they would say to make vaccines optional. This, of course, would be a disaster for society as it would no doubt lead to an erosion of vaccine rates and an increase in vaccine-preventable diseases.
Regarding the second part of your question, I would argue that vaccines are the safest, best tested medical products that we put into our bodies. So the issue isn't one of data (plenty of data are available to address these concerns). The issue is one of making those data compelling. And that is where I think we have fallen short. We have to be able to grab the attention of the media and the public with the data that can exonerate vaccine fears.