I *was* that anti-vax mom, I am embarrased to say. It took a looong time for me to realize how wrong I was. My kids' pediatrician never once questioned me on my decision to not vaccinate. I wish they had; maybe I would have come around sooner. Maybe I wouldn't have spent so many years spreading this dangerous misinformation to others. Awful, just awful.
I don't work in pediatrics. Most commonly I deal with this related to gardasil and flu- for some reason people are especially dubious about these two.
If someone is vaccine hesitant, ask them why in a nice and conversational manner. If possible, explain how vaccines work, address their concerns with basic facts, and don't try to get them to agree. Just say the info and move on. I usually say something like "Okay its totally your decision, but you deserve to know the info so I'm just going to run through it with you, to make sure you don't have any other questions." No scare tactics, zero pressure or expectation. Top priority is maintaining that therapeutic relationship.
It takes slow exposure over time. You likely will not change someone's mind in the moment, but you've planted a seed.