As has been rumbled about 'round the forums there is a need for for gold sinks for the large amounts of EC coming into the game. One such proposal has been that Feet Starbases--in whatever form they take--will act as gold sinks. This sounds like a good idea, except for one problem.
What happens if a fleet breaks up? In the case of bank items or credits, this could be split up, but what happens when you have three of more factions each insisting that the BattleStar Awesometown is theirs?
One form of EC sink that the Fleet Starbases can have is a weekly or monthly maintenance fee and you can buy equipment for the starbases that adds some additional functionality to it (example: craft special items that can only be obtainable from there that are fleet bound) that also increases the maintenance fee. Ideas taken from the clan hall system in Lineage2.
As for the problem you described.
Starbase is bound to the Fleet, so if the Fleet brakes up then it's the person that is in charge of the original fleet that gets it unless he/she transfer the fleet to one of the others. It's the same in almost every MMO that has something that a fleet can own and it's upto the players in that fleet to come to an agreement, not the company behind the game.