I don't know if an influence starbase qould have any effect on a planet. I had the distinct impression that the industrial bonuses worked fine, and I'm certain military assistance modules work as well.
Influence on the other hand works differently (AFAIK). The planet generates an amount of influence which determines your "borders" on the map, and if you have influence starbases then the your influence int the sector should increase, which means that your borders can move a bit farther, if your influence in the edge parsecs in now bigger than your enemy's.
Hmm I find myself having a lot of trouble to explain influence with words...