Enthusiast weighing in here, but I think it has a lot to do with a painting. In a roundabout way, it seems to me that the contemporary interest in the crossing of the alps stems from the Napoleonic fascination with Rome, particularly the imperial aspect of it. It was quite the popular theme in post-revolution France:

The stoic virtues of Republican Rome were upheld as standards not merely for the arts but also for political behaviour and private morality. Conventionels saw themselves as antique heroes. Children were named after Brutus, Solon and Lycurgus. The festivals of the Revolution were staged by David as antique rituals. Even the chairs in which the committee of Salut Publique sat were made on antique models devised by David. ...In fact Neo-classicism became fashionable. (Honour, Hugh (1977) [1968]. Neo-classicism. Style and Civilisation ).

You can still see traces of the post-revolution neo-classicism in Paris today, for instance, in the famous Arc de Triomphe that memorialized those that died for the revolution and, later, all of France.

Napoleon capitalized on the association of the revolution with the classical era through various propaganda techniques. One of them was the utilization of the painter Jacques-Louis David.

David was already famous for his work depicting revolutionary Jacobin martyr-figure Marat, as well as his recurring depictions of classical, dramatically renditioned scenes (see: Intervention of the Sabine Woman, Oath of the Horatii). Based off his reputation Napoleon commissioned him to paint a depiction of him "crossing" the alps, in what would become his most famous portrait, or painting for that matter.

The portrait, widely reproduced and displayed at the Louvre and in the palace at the time, was very clearly meant as propaganda that would draw parallels between Napoleon's empire and the great Classical empires. I'll let the National Gallery of Victoria explain how David did this, because it's better than I can do:

Inscribed into the rocks on which the horse rears are the names Karolus Magnus and Hannibal. Bonaparte is newly inscribed alongside them. Karolus Magnus is the Latin for Charles the Great, also known as Charlemagne, King of the Franks and Emperor of the Romans, who created an empire that spanned Central and Western Europe. During his reign he conquered Italy and was named Holy Roman Emperor by Pope Leo III. Hannibal was a Carthaginian military commander who led his forces across the Alps to conquer Italy.

Acutely aware of the power of the image, Napoleon commissioned David to paint him as a successor to the great empire-builders of history. It was one of many images that established the persona of Napoleon as a heroic and extraordinary leader of men. ([Source[(http://www.ngv.vic.gov.au/napoleon/art-and-design/A-Closer-Look-Jacques-Louis-David))

There were several copies of the portrait existing that ensured it would become one of the most iconic images of Napoleon, even while he was still alive. And in an increasingly globalized world, the news of Napoleon's crossing of the Alps, and copies of the painting itself, were seen the world over.

Essentially, at least by my theory, the Crossing of the Alps may have entered the public conscience (as opposed to that of the private military conscience) with Napoleon's attempt and this subsequent painting, or at least they cemented it into most people's understanding of history.. Of course, I could be entirely wrong, and perhaps the Alps-crossing was just as famous before Napoleon's attempt as after.

It was a well known event from Roman History. A "proper" education of those who could afford education from the 16th century onwards would have involved the study of Classics, including all of the known Greek and Roman writers, and they would certainly read about Hannibal, and would have known about his exploits with his elephants. Buonaparte and his propagandist, David, certainly deliberately fostered the legend, linking him, as has been noted, to Charlemagne as well. There are many representations of this event from well before Buonaparte's "exploit", which was more accurately portrayed here http://www.liverpoolmuseums.org.uk/picture-of-month/showlarge.aspx?id=282. This one was by Jacopo Amigoni, painted in the first half of the 18th Century http://www.reversespins.com/alps2.jpg This one is by Poussin, http://artintheblood.typepad.com/art_history_today/2013/09/the-elephant-in-the-room.html painted in the 1620's. This 18th Century illustrated edition of Polybius has lots of images of them http://spenceralley.blogspot.co.uk/2009/07/hannibals-elephants.html