Influence starbases should be placed near your borders to push them out. |
Please correct me if I'm wrong, but an Influence Starbase (IS) can serve three purposes in extending your overall influence:
1. Exactly as HEEGZ has said - Placed on your borders, they increase your physical range of influence.
2. Influence is Relative - It applies to your empire as a whole. By building an IS anywhere, its output contributes to your entire sphere of influence (i.e., its bonuses apply to your overall influence progression -- the "growth" of your overall area of influence).
3. To Piss Off Your Enemies! In lower difficulty levels you can build these near an AI planet to try to decrease his hold on said planet. If the race doesn't destroy the IS, you have a decent chance of taking the planet over (if it currently has a low morale -- 40% or less). This can take time, though (not too sure how long, but it's dependent on the current morale level of the planet).
If I'm wrong on any of these issues, feel free to clarify (because if I am, I've been playing wrong so far!!

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