Starbase and Mining Resources Question

No, you can't mine the resources at the same time as them, it's first come, first served. The only peaceful way I know of to get the resources permanently is to trade for the starbase with the other race using diplomacy (Alternatively you could just trade for the resources themselves, although this will only give you the resources temporarily). I don't think influence will flip starbases, only planets.

Thanks. I guessed as much. I feel like if another race has a starbase in your zone of control, there ought to be a way to gain it back from the other race. Perhaps a reduced price in trade due to the player's influence in the area in which the starbase exists.

There's always Gunboat Diplomacy.

In my experience, AIs normally won't put starbases directly inside your borders anymore unless they really don't like you, and if your borders grew around the starbases over time, trading for them is usually your only peaceful option unless you went Benevolent, in which case you can use Prominence III to culture flip them all (a one-time action). Unfortunately for peace-lovers out there, the proximity of starbases to one's borders often turn into border disputes, since the player near the resources considers the resources a part of their ZOC, but the player who claimed the resources was the first one to plant their flag and start mining it, declaring a galactic version of finders-keepers. They logically won't give you a discount in the negotiations, because they want those resources too. Their proximity to you is, to them, "a meaningless coincidence."

So, it's actually supposed to make you mad when the aliens start claiming your shit. It's how wars start, and as far as I can tell, that's how they're supposed to start.