The best use is to place it touching one, if possible two, of the planets you want to flip. If they are all close enough, flipping the first one generally causes a domino effect.
Influence bases generate their own influence, they do not affect the influence output of your planets. Their effect also decreases much faster than that from planets, so closer is much better. The ring is essentially worthless in judging the effective radius of the base - put them in contact with the planet you want to flip for optimal results.