I mostly use them for protecting relics or ascension crystals. Haven't seem too much of a bonus to using them yet. Near enemy planets I like to drop a culture starbase (preferably with lots of planets in ZoC) and watch the culture flips happen. Especially the enemy I am silently at war with. Hope there are more users for them though. Would be nice to find a good use for them.

I tend to put them as close to the planet as possible, or if there is a cluster of worlds in the center of those. The closer the better as you want to flip them quickly and not have to wait for your own culture to build up which can take awhile. Also helps if the enemy world isn't as developed so there is less resistance.

Military starbases are good for defense. I play with the vigilant technologies so I have defensive economic starbases around my colonies and then an outer line of military starbases around those. I play a defensive game so my first 6 or 7 citizens are usually administrators for starbase construction. When my defense is set, my fleets can move back and forth between my protected zone without using hardly any movement points and have massive offensive and defensive perks while penalizing the attack, defense, and movement of enemy fleets.

I play godlike with 60 AI, if I cant get my starbases up it's over, a lot of the time it still is anyway.