To N3rull-Dude, lighten up. I'd like to remind everyone, before a flame war starts on a POTENTIALLY USEFUL thread, that it's a game. It's not your life, so get over the egos and the hostility. Treat it like a problem, not someone trying to make your life miserable.
Done preaching now. Okay, the problem, to restate, is that you have a 23k star base, probably all three weapons upgrades, and ? frontal shields in a system, that, for whatever reason, you HAVE TO HAVE. Now, as the TEC or the Advent, the answer is obvious, anti-structure frigates and bombers. Whee, obvious solution. As the Vasari, however, you have no anti-structure frigate and it takes too many bombers to really be worthwhile. So, let's go with both scenarios.
1.) You can ignore it
Ignoring it, if possible, is probably a good answer. Just go kick his butt somewhere else. Unfortunately, as N3rull pointed out, sometimes this is a bad idea, if your core worlds are one jumpr from this Starbase. As the Vasari, you have Phasegates. I'm assuming this is late game, and you can afford 1, phase gates, 2, a defensive fleet, and 3, four or five starbases. Just put a starbase and phase gate around the vulnerable worlds, or combine them with the phase gate tech for the starbase, and get a defensive fleet. Juicy world problem solved. Your fleet is also one jump away from any of your vulnerable worlds. Meanwhile, you can go annihilate him elsewhere.
2.) You can't ignore it
A few possibilities. One, you have your kostura cannon, which you then shoot him with. His fleet is in bad shape and probably retreats when you jump in, leaving you free to take out the starbase. Alternatively, you can make a fleet of Assailants, Enforcers, Overseers, and Transporters, heavy on the Overseers, with, of course, an Egg, and just duke it out with the damn thing. Not being able to ignore it is much harder, of course, but if it's a choke point, then do the phase gate/starbase thing, and wait until you can afford a cannon, possibly raiding his worlds that are a dozen jumps away with a light, fast force, maybe Assailants/Sentinels with the Marauder for quick escapes (Yes, I know, it's junk, but it does speed up the fleet). You could also jump in four or five Ruiners and lay mines all over the place. That's more annoying than it is effective, but it could be enough to give your fleet the edge it needs. Mostly, the 'You can't ignore it' tactics boil down to stalling and waging a war of attrition. The other thing you can do with the cannon, which I can't believe I forgot until now, is to shoot the assumed Kostura cannon at one of his core worlds, jump your fleet there, and do as much damage as you like.
For both of these strategies, the key is to remember the Vasari's advatadge in remaing mobile, and to use it against the stationary defenses. Remember, averything is situational, so don't plan on there being a 'By the book' offense for every situation.
Alright, I think I covered all the points that were made, by both sides. If I missed something, point it out, and I'll be happy to think about it and post a reply. I realize that my tactics for not ignoring it are a bit lacking, but honestly, there's not as much you can do, but there is ALWAYS an option, or else you've already lost.