I don't know if other people's games have this trait, but most of my games involve characters of some pretty varied backgrounds - Starfleet officers, Klingon warriors, Romulan agents, and smugglers. Getting them all together presents something of a challenge.
Assuming Starfleet as the "default" background, there are three setting types that I have used or considered...
Starbase/Outpost - Set in the model of Deep Space Nine, where the Starfleet characters are in positions of authority but not in total control, like on a starship. Gives the non-Starfleet characters something of an advantage in dealing with certain NPCs (i.e. Kira with the Bajorans, Garak with Cardassians, etc.)
Little/Medium Starship - For some reason a Starfleet vessel has a few "alien" crewmembers. This is the case as seen in Voyager (with Neelix, Seven of Nine) and Enterprise (Phlox, T'Pol). The smaller ship size means a Starfleet crew would have a greater need of such characters (7 of 9 being a backup "chief engineer" it often seemed). This setting allows the "alien" characters backgrounds to be explored from the perspective of the Federation and keeps the Starfleet characters in a familiar setting.
Big Ship - This would be the case aboard a Galaxy-class ship or something similar. Think of cases like if Worf's brother Kurn stayed aboard as first officer. In such a case the setting resembles that of a small or medium ship. But a big ship also offers the possibility of a joint mission. Perhaps a Federation ship needs to travel through Romulan space (or in space adjacent to it) or a band of smugglers has information needed by the Federation but, by itself, worthless to the smugglers. Perhaps a group of Cardassians is assigned to a Galaxy-class vessel to help keep order in the ruins of Cardassian space post-Dominion War. Whatever the reason, you have something of a mobile starbase, with enclaves of the aliens, giving a microcosm of the alien culture.