With previous (well-established) vaccines, even in those instances, you find the same anomaly where there's no partisan line to be found.
When you encounter an anti-vaxxer, there's a 50/50 shot you're either talking to a far-right anti-authority type who thinks that that "the government is trying to poison people for population control", or you could be talking to a far-left person who's into "everything has to be natural from mother earth, here, try these healing crystals and essential oils"
Sort of the same dynamic that exists with the topic of GMOs.
The reason people can come from different sides of the political aisle on this is that it is a question that all must face regarding real risks to their child or themselves, and it cuts across all groups. Most parents considering whether to vaccinate their child are not looking to be revolutionaries or reveal secret plots. They just want to do what is best for their child, but may not be sure what option is best.
The way to convince them is not to insist they are backwards or superstitious, or to categorize them as this type of thinker, or that, or compare them to those who hold a notion you clearly have disdain for, but to show them the data on prevalence of harm from the disease, vs prevalence of harm from the vaccine so they understand the risk they are taking, and why.
However, when trying to convince those who are hesitant I have seen people take an approach that is more about scaring people, or pressuring, or belittling those with concerns as wrong-thinkers, rather than presenting data. If it is a slam dunk on the science, then publish the risks on both sides. If it is safe and effective the data shows that.
In my experience public health campaigns and physicians often only want to discuss the risk posed by the disease. Any questions about the risk from vaccination are downplayed or dismissed.
It only takes the person knowing a few examples where a vaccine caused serious harm for them to question whether to go forward with a vaccine. The solution is to be upfront about risks on both sides, but demonstrating that the risk from the vaccine is small compared to the risk from the disease.
If you only show the risks on one side, but the person is aware of risks on the other side they feel you are not being straightforward. If doctors won't engage in discussing the manufacturer label side effects and how common they are, a skeptical person might wonder why that is the case. If we are asking people to make data driven decisions, we have to provide data. It should be ready to be presented to those who have a concern.
And when data is not provided upfront then the person feels the need to go digging to find it, and often winds up obtaining it from internet based advocacy groups. Depending on the site or group that may mean they get a skewed picture, or an exaggerated version of the harms from vaccines.
If the internet site tells you examples of harms, but your doctor refuses to discuss it, which appears to be giving it to you straight? Both may be giving a skewed picture, but one is willing to talk about harms and the other is not. If it is data driven then we can't be afraid of spelling out the data, and the relative risk on each side.
The data is out there from various sources about how deadly diseases are, how prevalent harms are from vaccines, etc. And to me I found that the case for vaccines for myself or my children are usually strong. But I didn't appreciate that I had to go find the data because they wanted me to accept it on the basis of slogans, or authority, or pressure, rather than just spelling out the risk vs. reward to start with.
A parent doesn't necessarily need to know all the science of how vaccines work to read x number of people experienced life-altering outcomes from this disease, vs. y number of people experienced life-altering outcomes from this vaccine. If they know that there is a real but very small risk from the vaccine, vs. a real and larger risk from the disease, then the decision becomes clearer. Characterizing the data on either side by glossing over it doesn't convince people.