I had a few games where i focused only on building influence starbases. I play at the "tough" level. I think there is something wrong with the way it is done. I tried following the same playing scheme everytime, and it worked every time.
First of all, i played on large galaxy with all the other things randomized (6 opponents set at intelligents). What i did : expand (not very much cause in my last games the habitable planets were uncommon), then rush my closest neighbor with medium ships and a few transports. I conquer my closest neighbor. After this, i only develop yellow techs : trade, influence techs, etc... Then i watch the biggest civillization out there : the most powerful. Often, it is the Arceans. I ONLY trade with them : i send all my freighters to them. They love me. Finally i conclude an alliance with them.
The strange thing is that while im really behind military speaking, the other civs don't declare war. I have a few ships on each of my planets, but nothing really impressive. When i have researched most of the yellow techs, i only build constructors. I pick a civilization, place a rally point near its homeworld, construct influence base, and spam constructors. I build every influence module i can. The planets flip one by one. Of course, i get the "Hey, you are building an influence starbases near my worlds : go away or i will crush you" sentence ; but they don't have the time to do anything about it. They just watch as their planets flip.
They SHOULD declare war ; if only for one turn, it would allow them to destroy my influence starbases and then, maybe, beg for peace... But no : they just watch. And they die slowly, one by one. I conquer the whole galaxy this way, and, being allied with the major civilization out there, i soon win a diplomatic victory.
What is strange is that all the civs are at least "neutral" with me, even while im spamming 3 or 4 influence starbases near their homeworld ! (btw - do their effect stack?)! They all like me, but im killing them !
If i was facing a human player, he would instantly declare war, destroy my unprotected starbases, and then ask for peace ; at least he would do something. But the AIs are only watching it happen without doing anything.
Maybe this needs tweaking? What are your experiences with influence victories?