Alien Influence bases

I would probably destroy them. Then if you real don't want war try and talk your way out of it afterwards.

try trading tech or money for their starbase

It does seem odd that you can plop down starbases in someone else's territory with no consequences. In the end though, I guess the consequence is that you're goading someone into a war. Give them what they want and don't stop at the starbase!

So you mean that if AI makes his bases near my plantets it is an act of agression?

What would do AI if I build few starbases in his system?

The way of countering it is building one of your own to overlap there's (like build one that covers also your two planets) or build some influence improvment on your planet (embassy).... or research tech that give you influence bonus. I
nfluence is a powerfull weapon, one thing I would like to see is being able to taxe them or prevent them to build on my influence zone (other than by the UP vote, by the way who is making those propositions? I never had a chance to choose one). Of course for balance purpose you will need a down side to this approach (like a good minus to your diplomacy with this civ), this could then lead to a real "influence war".

I do not posses the game yet, but how can enemy (alien) ships cross your borders and place entertainment starbases near your planets? Or maybe he does simple 'influence bombs' (like in Civ 4), so that he builds starbase in HIS territory, but eventually he will broaden his influence on your territory.

In the end, there is always one soultion - WAR!

There is no hard borders as in civ4, it's simply your area of influence, where your culture is dominant. So the IA can build base right in the middle of your system, you can do the same.
Influence bomb is not the right term, it slow to upgrade a influence starbase to have it be really effiecient and cost a lot of money. Also if you do that the IA can pest you about it, she event treaten you to have you destroy it (but she never attacked me for that, for a reason, I always make sure my neighbours are busy somewhere else )

How about when your ally places an influence starbase next to your home planet? It happened to me in the DL campaign, and seriously pissed me off. They evey upgraded the darn thing.

Just pay off another civ to go to war with who ever is creepin in your hood, send some ships to their bases, and then sell/give the ships to the civ you paid to go to war with them

Yeah, the Ally underhanded tactic of influence bombs IS a bit annoying, and I do hope Sdock looks into altering this behavior.To me, it seems MUCH more appropriate for them to build econ bases, because chances are since they are your ally, you have a nice trade setup with them.

If you can't build hapiness/morale things on the planet, you are left with little choice: destroy them.
As suggested, do it yourself, or pay someone off to do it for you.

Also you CAN offer to buy them, but my experience has been that the AI doesn't like to barter over THEIR starbases unless it's part of a peace treaty. And when they DO barter, they are usually at a hefty price. It is so swell when you get them to hand it over, only to see them pop another up in the same sector RIGHT after you thought you got rid of the problem!!!

Another thing to try is to copy them. Build an inf base in their sector, build it up to equal strength and after a few turns, offer to even trade them.