Perhaps I am missing something about the claim, but as far as I can tell it is ludicrous on its face. The carte de visite, a small, photographic portrait card, was invented in 1854 and immediately became very popular. Abraham Lincoln's distinctive look, for example, was well-known by Americans of his time, largely through the distribution of cartes de visite (also through engravings in newspapers and other mass media). The prevalence of such mass-market photography only grew throughout the course of the nineteenth century and into the twentieth. Hell, there is a very good book, called Vision, Race, and Modernity: A Visual Economy of the Andean Image World that explores how cartes de visite were immensely popular in the nineteenth century among indigenous and mestizo peoples in Chile, Bolivia, and Peru, which gives an idea of how widespread this technology was.

So, to answer your questions more directly, the average American would have known what the President and other celebrities looked like as far back as the 1850s or 1860s, and definitely in the early twentieth century. Secondly, likenesses had been included in newspapers essentially as long. At first, this was accomplished through a process known as wood engraving, when a photograph would be translated by an artist into a wood block, which would then be printed in the newspaper. In the 1880s halftone printing became commercially feasible, and photographs could be printed more directly into newspapers. The Benday dot, a similar method for reproducing photographs in print, was also invented around this time.

I can't think of a technology that might have enabled further awareness of what the President looked like around the time that Taft was elected - perhaps the newsreel film. I'd be interested to hear theories, or to read a source where you heard about this.