I understand your fear. I hear Play Doh remedies the situation somewhat. Factophobia is a huge mental condition.
Fact is, these immigrants were put through a strict vetting process and medical exams. Thousands died at Ellis Island due to disease. Many were quarantined with TB, scarlet fever, etc...People such as yourself think they should simply pass through a turnstile, climb a fence, or wade across a river. Why did you have your kids inoculated? Would that be a precautionary measure? Let's not take precaution when protecting an entire country. That would be dumb huh?