I saw on the latest episode of QI that Samuel F. B. Morse (of Morse Code fame) was a painter who would make paintings of the exhibits at the Louvre for people who were unable to visit. How many people would have known what the Mona Lisa would have looked like at the time? I'd assume that typically only people who had traveled to Paris would have seen it.
It depends what you mean by "most people." Just during the 16th century various copies were made of the painting (e.g. the Prado Mona Lisa). Nonetheless, throughout early modernity, the painting was not the popular, iconic image it is today. It was in the 19th century that the Mona Lisa slowly began gaining its current status.
It could be argued that the image that made the Mona Lisa widely recognizable was an engraving created by Luigi Calamatta in 1857, and which took him 20+ year to complete (Carl Goldstein, Print Culture in Early Modern France, 38; Stephen Bann, Distinguished Images: Prints in the Visual Economy of Nineteenth-Century France, chapter 1.).