The most important thing is to figure out their agenda.
The anti-vax crowd has a shit-ton of overlap with the home birth crowd has a shit-ton of overlap with the "western medicine is bad" crowd. So is it ideologically-driven or ignorance driven? My wife had a partner for a year or two who straight up said she had doubts about germ theory. The writing was pretty much on the wall there - no reason to check your expectant moms for Group B Strep if you don't really believe in, like, the accepted model of disease. She'd toe the line so as not to jeopardize her license but it was lip-service.
The ideologically-driven folx don't really give a fuck what the data says. These are people who treat their kids with homeopathy because Western medicine is somehow offensive to them. They're also the ones who tend to show up in a panic when homeopathy doesn't "cure" something but does, in fact, allow it to get worse. They'll often throw up their hands at that point.
The ignorance-driven folx looked up "vaccines" on the Internet. It's better than it was, but even if you google it right now, the top three news articles are "jenny McCarthy is anti-vaccine" (who cares) "vaccines are good" (USA Today) and "vaccines are a choice" (USA Today). If you let autofill guide your way, you get "vaccine dangers" and "vaccines cause autism" before you've typed V-A-C-C. Let's say you just go for "vaccines" and want some actual data - Yay! There's the CDC's page. Except pretend you haven't slept more than four hours in a row for eighteen months, you've got a screaming baby in the high chair next to you and you just want to know if your baby is going to be okay when you take her to her checkup and now go look at CDC.gov.
Know how all the ads say "ask your doctor?" The system is designed for you, the patient, to have a trusting, friendly relationship with YOUR DOCTOR. In an era of HMOs, PPOs and Obamacare, you don't even remember who that is. He or she probably had a thick accent, spent a lot less time with you than the phlebotomist, and operated out of an annex in a mini-mall. As a system, we've done everything we can to make patients feel powerless and practitioners like tiny cogs in a giant system. Into this, inject "trust."
So the "authority" becomes the person who can most effectively communicate.
My wife prints out a fresh CDC vital statistics report for every class. She's read most of the books available on vaccines on both sides of the debate... and will freely acknowledge and remind parents that the ones in favor of vaccination have the facts on their side. At the same time, if you look at the vaccination schedule over time you see it ramping up like a mutherfucker in the '90s - your kids are on track to receive about three times as many vaccinations as you did. This prompts the argument about whether or not they're all necessary. Which usually leads to a debate about varicella. Keep going down that route and you can blow the FDA out of the water with Vioxx but you have to remind the parent that FDA does not equal CDC.
In the end, the parents that were on the fence about vaccination tend to go for full-bore vaccination and the ones that were skeptical of vaccination tend to go for a modified schedule. The ones that were adamantly anti-vax - the ones that show up at all - get a bunch of information about how to treat their kids like Bubble Boy whenever there's anything anywhere around them. 'cuz that's the thing - raising unvaccinated kids is a stone-cold bitch. My kid was running a little late (she had an onslaught of ear infections that made our pediatrician wave her off for a couple months - I have no problem with vaccines) and it totally impacted our ability to drag her around.