Influence starbases don't increase the influence of any planets in the ring, it only increases its own influence. You can place one deep in an AI's territory to open a "bubble" of your territory, even if you don't own a planet anywhere nearby.

Also, the influence from a base drops off faster than it does with a planet. Planet incluence seems to drop off nearly linearly with distance, while base influence drops at closer to distance squared. The important thing to take from this is that an influence base is most effective when built touching the planet you want to influence.