If those "other people's" children are vaccinated then what is the problem?
The problem is the vaccine mentality. I grew up before there was any vaccine. We all tried to get the measles or chicken pox just to get out of going to school. Now the disease is a deadly one. Bull S—t. They never state how MANY ARE GETTING THIS DISEASE THAT HAVE BEEN VACCINATED. The vaccine is not that good. How many die or are crippled due to the vaccine?
There are several problems.
Infants might not be vaccinated at all because of their age. Except for HepB, they don't receive any vaccines until 2 months of age, and through the age of about 6 years, they must keep receiving boosters in order to fully train their immune systems to protect them from those pathogens. With their immature immune systems, they are especially vulnerable to severe life-threatening illness from diseases that might only be an inconvenience to older children and adults.
Vaccines are not 100% effective. Because of differences in immune systems, people respond differently to different vaccines. A few might be partially protected, others not protected at all. You don't know who those people are until some unvaccinated idiot comes around and spreads the disease. In order to make sure that everyone benefits from vaccination, about 95% of the population needs to be vaccinated.
There is also the selfish attitude displayed by the anti-vaxxers. They exploit the fact that your vaccinated kids are very unlikely to spread any diseases to their spawn. But they have no problem with their spawn giving deadly diseases to your babies who are too young to receive a full vaccine series.