Stars can have more than 1 Starbase? Never noticed that.
What exactly are the rules for starbases? Do other players' starbases effect anything I do?
Well, yes, on a star they will affect hull when phasejumping out, making them great for weakening a fleet before they enter your primary fortification.
If you get it right, with the advent, for example, you can place four starbases together near the most frequently used jump-position, upgrade them all to max defense - all of them with 4 squadron-upgrades. Alternate the remaining points between disruption and meteor strikes. Usually I have two in front that have 1 weapon upgrade, 3 def upgrade and four strike craft upgrade. The farther two have the meteor strike capability and the disorient ability.
With this, any fleet cut in their path will be cut down to cinders.
Even if your opponent avoids them entirely, you can send out up to 80 bombers (if you're so inclined) to decimate their caps.
Usually I go 30 fighters (to ward of enemy bombers) and 50 bombers. 50 Bombers can do a shitload of damage to a fleet before it manages to jump out of the stars gravity well. Particularly if you micro-manage a bit and focus their fire properly. The Vasari cap that can phase out its hull doesn't do that when it's moving. Not in my experience, anyway.