They didn’t get much traction.
Early on, they got plenty. It was just restricted to the tight media networks of the surburban UK and US. Most of the work eliminating smallpox was international - Latin America, Southeast Asia, and Africa. By the time you had mass media inundating the public with glowing coverage of these efforts, mandatory smallpox vaccinations had ended and the enthusiasm for both promoting and resisting smallpox had waned. Virtually nobody gets smallpox shots in the modern day. But, more critically, mandatory vaccinations have fallen away in favor of "opt-out" rules for school attendance and military enlistment.
They used to be a bunch of wackjobs in the corner
Really depended on which corner you were standing in. You had lots of crunchy suburban granola moms forgoing measles vaccines for their kids back in the 90s - a thing you could get away with largely because measles outbreaks had become so rare. Then California got hit with a number of nasty measles outbreaks and the media got into the practice of calling anti-vaxers wackjobs. But this, in turn, created a market for right-wing contrarianism to say they were against Big Government California Mandates (despite saying they were against stupid hippie parents not vaxing their kids ten years earlier).
There's an ebb and flow to this. Anti-vax sentiment was not always the norm, but neither was it always taboo. A lot of it boiled down to people being contrary on opposite ends of the liberal/conservative dichotomy.