Today’s parents worry constantly about the kids: What are they doing, seeing, saying, eating? Is that shampoo safe? Are they watching too much television? Thanks to antivaccine campaigners, add measles to the list.

In 2000, doctors were celebrating that measles essentially had been eradicated in the U.S. Even people who could not safely receive the vaccine, such as newborns, were protected by what’s known as “herd immunity.” When a large portion of a population is immunized, there are few potential carriers to spread the disease,...