Sure the already anti-vaccination crowd will have more reasons to be peeved and more talking points to scream about. But would it mean new people would turn against vaccines because they hear "DNA"?
It seems to me like the big divide in the general population is over people who are open to their conceptions of nature be challenged and corrected (this whole pandemic has been a mini science course for a lot of people: exponential growth, conditional probability, mRNAs, to name a few) and those who just are not. Any other pre-existing opinion/ideology ("I just don't do vaccines") is probably unaffected by technical details.