I just did an experiment. I noted my influence produced at my homeworld by moving the pointer over the influence score at planet view. I contructed 2 constuctors and built an influence starbase with an influence module added. I ended turn. Went back to planet view to note score again. Nothing changed.
So... Influence starbases due not boost planet influence. They either boost a "squares" influence or produce their own influence or both.
Thus,in general, they are best used by building them to have opponents planets in the area of effect; as stated by some other people.
-Wade