Depends on what language. The Late Republic had at least a dozen other languages than Greek and Latin, some of which died out as a written language. Most people probably learned Latin in addition to whatever local language was prevalent.

If you want Rome specifically that's harder. Harris' overall estimate for literacy in the Western empire is about 5-10% - elites were literate, (women perhaps even at the same level as men) and if we assume that Rome had a few more elites, then you might push it a bit higher, but Latin was still spreading in Italy during the 1st century at the expense of Oscan and Etruscan which both probably died out by the 1st century AD.

There appears to be a slight decline by the end of the Empire, as the cities declined, but it appears to not be very much of a decline as literacy wasn't very high to begin with.