I recently saw a story of a 19 year old who just rowed across the Atlantic by himself in under two months - about 60 miles per day.
With sails and currents, other could probably do it much easier.
Traces of cocaine and tobacco have been found in Egyptian mummies, both sourced only in the Western hemisphere. Trans-Atlantic trade has been going on for thousands of years.
Cruisers numbering hundreds if not thousands per yer make the trek across the Atlantic without much fanfare. I've read stories of sailors on 26' ketches making it from Cape Verde to the eastern Caribbean. Most cruisers over 38' are comfortable enough to provision and cross in less than a couple weeks, depending on winds.
The difference between then and now is that we know the boundaries of the Atlantic thanks to cartography and satellites. Back then, to them, it was just open ocean as far as the imagination could go. As long as they stayed out of the doldrums and sailed right west, they could conceivably make it on personal watercraft. Would take some serious stones to want to do that though.