There are two basic ways to approach this.
One is the "keep everything as close to real history as possible" approach. This is the approach Greg Cox's
The Eugenics Wars novels back in 2001 and 2002 took. In these books, Khan is depicted as having seized control of much of Asia covertly, using blackmail and coercion to keep the national governments taking orders from him while keeping them in charge on paper; Augments were also depicted as leading various nationalist movements, such as the warlords who ascended to power in the Balkans when Yugoslavia broke up. In these books, the American public largely remained unaware of the role of the Augments in world affairs, due to a combination of ethnocentrism and ignorance; it was only after the Eugenics Wars that the Augments' role in society came to be understood.
The other way to do it all, of course, is simply to presuppose that
Star Trek's history diverges more completely than this, much earlier, and to depict the Eugenics Wars as a World War-level conflict in which the roles of the Augments are well-known by the public. This is the approach taken by the current
Star Trek: Khan comic series, and by last year's
Federation: The First 150 Years book by David Goodman.
ETA: The smartest thing
Star Trek Into Darkness did was avoid the issue one way or the other.
