Bombers or ogrovs shouls work, but beware that if the advent starbase has meteor control things might get messy... have some cielos and hohos near your anti structure ships to help.

Normally, i would say just go around them, but on such a small map and with a base at every planet you are going to have to kill them eventually. If you have a lot of resources, up fleet supply and use the TEC economy to outproduce your opponent, MB and GRG are your friends...

These are all my tricks against starbases:

Jerrasul Evacuator with Nano Disassemblers:  Excellent against starbase with no more than 1 health upgrade.  Any stronger and it will need backup.

Bombers and more bombers:  the most practical way to take down a starbase.  They work against any starbase regardless of its upgrades, also perform well in fleet battles.  Definitely your best option agianst heavily upgraded starbases in the late game.  Particularly useful against Vasari, just make sure to keep your frigates and cruisers out of range of its guns.

Ogrovs and Adjudicators:  good against starbases early on, relatively inexpensive for their strong performance in battle, but useless in a fleet battle.  Use when you can't afford to bring out loads of bombers, but need to take out a few major starbases.  DO NOT use against Vasari. 

Vasari Starbase:  expensive and not very quickly to act, but it will break TEC and Advent starbases given enough time, and you'lre left with a powerful defensive installation afterwards.  A good option you're trying to take control of a key choke and will be starbasing it anyways.

Guardians and Progen:  Use the guardian and progenitor mothership to keep your units (likely illuminators) alive as they attack the capital ship.  Be very mindful of any uncapped area of effect ability or some of the Vasari starbase's defensive capabilities.  If your progen or guardians run out of juice, you're in trouble!

Another thought on that topic:

Since the AI is Advent it is likely that beside the Starbase he will have some Hangar Defenses packed up in his grav wells. I recommend to stay out of the SB's range and take those out first (and the constructor ships) as the fighters might shred your bombers. Keep the bombers docked until the Hangar Defenses are destroyed or mix in some Gardas to deal with enemy fighters.

Then of course Bombers are the way to deal with the starbase.

Thanks for all the tips guys!

*EDIT*

Just completed the game with ease. All I had to do was make my fleet bigger, that's all. He surprised me sometimes by showing up with a fleet of 2 caps and illums out of nowhere, but he never managed to penetrate through my minefield back up with turrets and my Starbase.

Once I got my fleet up with Ogrovs, Flack Frigates, Carriers, Kodiaks, a KOL at lvl 8 and a Marza at lvl 5, he was pretty much beaten. My Marza grew in the process and eventually got up to lvl 9. I first took out all structures and hangars while my bombers took care of the Starbase. At the last planet it took the bombers an eternity to take out that last Starbase with 13,5k hullpoints and 6k shield points, so I sent my whole fleet on it. My Marza pretty much took care of it with the Ogrovs and LRMs. Missile Barriage was incredibly effective on the Starbase, and I did not loose even 1 single ship fighting through 3 planets.

thanks for the tips on taking out enemy starbases.  I'm a newb to the game and I coming across my first game on Hard difficulty playing as TEC. 

I have a slight upper hand in control of the map (sumfink like 8 planets to enemy 6) - but the enemy has these BUFF starbases (like ~15,000 defense points! insane!) that can move around the planet!  I tried going toe-to-toe with one enemy starbase using my fleet and got decimated.  I almost lost the game when enemy counterattacked my planets after my foolish attack!

I will try the recommendations to outproduce my enemy.  I think I will try to "raid" his planets and decimate his infrastructure leaving the starbases for last.

How about this novalith cannon?  is it worth pursuing in research?  I may try it out for curiousity reasons.

best way is to go around,heroes always go around - thats how they stay alive.

If you just want to bombard a planet you've already conquered, Novalith is a very poor choice.  It will leave fallout for several minutes preventing you from colonizing, delaying those critical forward factories, broadcast centers, and repair bays.  Buy a new Marza if you need to speed up bombardment, use Novalith pairs to destroy planets deep in enemy territory to screw with his econ.