The headline here suggests yes. But the article gives no evidence on this, and instead spends most of the time giving evidence about the possible link between vaccines and side effects. Here’s the only suggestion of elite leadership I found:
Historically, Dr. Willoughby said, vaccine scares have caused vaccination rates to drop for three or four years, and have led to outbreaks of diseases that had previously been under control, like measles and whooping cough. Measles cases in the United States reached a 15-year high last spring, with more than 100 cases, most in people who had never been vaccinated.
Even if the Times had a plot showing a correlation between elite statements and vaccination rates, causality wouldn’t be shown. The basic problem is that elite rhetoric could anticipate, instead of echo, what the mass public thinks. This issue comes up a lot in public opinion research. Maybe vaccinations and these types of statements are a good way of testing this.