Basic was the common language of the Federacy, in which governmental business was conducted. It was also often the medium of academic teaching, especially where a university's fame in a certain field attracted students from many different worlds, such as the Anthropolgy Department of the University of Hyperion.
Basic was largely similar to English, which gave some advantage to native speakers of English, from the English-speaking parts of Earth and worlds settler from them. Still, this was not greatly resented by non-English speakers.
The use of English or English-derived languages as a lingua franca long predated the Federacy and even space flight itself, dating thousands of years back to the time when all humans were restricted to Earth. After this long passage of time, citizens of the Federacy worlds took it more or less for granted.
Moreover, an Earth-type planet is self-sufficient in most ways, providing a lot of room for a specific language and culture to develop - and many languages and cultures had more than one planet to themselves, for example Greek culture flourished in both New Thessaly and Alexander, as well as the original Greece on Earth.
As described by Stavros Monemvasios, New Thessalians who did not go off-planet or get a Federacy job could very well live their lives knowing no language but Greek. The same was true for many other languages having a planet or more than one to themselves.
Literary Note Edit
It is not explicitly mentioned that the Federacy Basic was derived from "Basic English"  - a simplified version of English created in the late 1920's and early 1930's by Charles Kay Ogden as an international auxiliary language, and as an aid for teaching English as a second language. However, this derivation seems probable given its name and its being used in the Federacy for precisely the purposes intended by Ogden, though developments over thousands of years and creation of a complex space-faring culture would presumably have changed it greatly from the original early 20th century version.