Military starbases: Try to place these inside important star systems or between you and your enemy. They give your ships in their area of effect all kinds of bonuses, like extra attack or defense stats, faster repair, faster ships, slower enemies... etc.
Economic starbases: Try to place these inside important starsystems also, especially if the planets need the bonuses... Also try to place them around major trade route intersection areas (your trade routes!).
Influence starbases: Put these by enemy asteriod miners (IF YOU PLAY DA) or by another race's planets (try to get the best planets you can!). Make sure you research far up the influence branch, and then keep installing the influenza modules.
ALSO: It is a very long and tedious job to upgrade starbase to their full potential.
the devs are still trying to figure out a way to make it less micro-managed.