Seventy-one percent of people think parents who do not vaccinate their children with MMR vaccine are being irresponsible, according to a ITV News/Com Res poll.
The figures come on the same day as the latest number of people affected by the Swansea measles outbreak is revealed.
The poll also shows that 83% of the British public consider the MMR triple vaccination to be a safe way to protect children against measles, mumps and rubella.
Other findings of the survey were as follows:
78% are worried that many parents appear not to have had their children vaccinated against measles, mumps and rubella.
85% of the British public support the campaign to give a million teenagers the MMR vaccine.
89% are aware of the recent measles outbreak in Swansea.
37% agree children not vaccinated with MMR should be refused admission to state schools, with 43% disagreeing.
42% agree that health authorities are not doing enough to protect children, 41% disagree.