Where do I place culture starbases?

Both, they give a bonus to the influence of a planet, but they also generate their own influence.

You can drop them right in the heart of 'enemy' space almost without challenge.  Playing against a Human such an act would probably start a war but against the AI it gets slightly irritated at worst, and such irritation is easy to mitigate with the odd trivial gift here and there.  The AI doesn't seem to consider culture warfare as a credible threat.

The first culture upgrade adds a flat +1 culture per turn. But the other upgrades just seem to add +% values, does the % of influence just effect planets or also the starbase itself. And can two starbases buff each other?

Thanks for the clarification guys, but doesn't that mean you end up with a metric fuckton of culture starbases if you're pursuing an influence victory? In my last game I had them everywhere, it was a nightmare keeping up with all of the constructors.

Sadly it can be a hassle, but sometimes necessary. On a tight cluster map with most the space empty I have to fill the void with bases to claim the 75% I need for victory.

I've got a follow up question. Does the effect of influence diminish based on distance? Let's say I have a planet that's generating a huge amount of influence, due to buildings and natural tile yields. However this colony is right at the heart of my huge culture borders, is it accomplishing anything?

It's generating culture, which is good for UFP meetings. If your empire is that large though, it's probably not doing much for expanding your borders.

You can hover your mouse over the map where cultural influence is present and see how much of yours / theirs is in place.  Everything radiates out from sources, slowly growing weaker as it has to overcome places where there is no / enemy culture.

Only mildly surprised that Culture spread isn't also yet another linear additive stacking effect.  It's a shame that decision didn't translate into some other areas of the game.