I think promoting the general welfare would more accurately prevent disease carrying people from entering the country.
And, that is the right answer!
People legally enter this country for all kinds of reasons every day. And since measles is contagious before symptoms appear, and it is one of the most contagious diseases known, you cannot possible prevent someone who is contagious from entering the country. As we are seeing with the Disneyland outbreak, it only takes one contagious person to encounter a pocket of anti-vaccine fanatics to cause a huge and expensive problem.
The rationale for mandatory vaccines is that you do not have the right to expose others to dangerous diseases, any more than you have the right to ignore traffic signals and barge across intersections, or to knock down your neighbor's house because it is blocking your view.
I really really wish it was only anti-vaxxer's kids who would bear the brunt of their parent's decision to leave them vulnerable to dangerous diseases. Unfortunately, it is kids of parents who very much want to protect their kids but can't, usually because their kids are too young to receive the vaccine. Babies who are too young to receive a vaccine are also the most likely to die if they catch a vaccine-preventable disease, and are therefore the most dependent on the herd immunity produced when everyone who is old enough is fully vaccinated.