And Europeans had no trouble crossing Europe.

A single smallpox laden explorer would have started a plague in the Americas and a single native American would likely have died of one of the many plagues in Europe that Europeans had grown up with (and still often died from).

The problem of safety is not always placed on the explorer, sometimes the explorer is the danger to his environment.

I wouldn't call a slow migration that has no written record on the same scale as the exploration of the new world.

>Indians had no trouble crisscrossing America

Well yeah there weren't any human infectious illness where... there aren't any humans.

This video explains the history of man kind's infectious illness very well https://www.youtube.com/watch?v=JEYh5WACqEk

The point is that entire classes of extremely common diseases in Europe basically didn't exist in all or most of the Americas, and many of those diseases spread very easily once even a handful of people had carried them across the ocean.