The only good answer Advent has to starbases is bombers. If there is no defending fleet 20 or so of them will make short work of an unupgraded starbase while the rest of your fleet can destroy extractors or start bombing the planet. Otherwise you just have to overwhelm them with the biggest battleball you can make.

Really they aren't as big of an impasse as you think. Without a fleet to defend them they are quite vulnerable, or you can just go around them quite easily. The real problem is that the AI doesn't know any of this, so your starbases are ridiculously effective against their attacks.

Starbases make it playable. With a starbase, you can fortify a frontline world without wasting fleet supply. That way, this place can shrug off big attacks with minimal fleet support, letting you defend two places at once. It lets you stay alive without having the 100% of your fleet to meet this guy's 100% of the fleet.

In my opinion, SBs make games manageable; they turn a massive microfest of raiding and shooing enemy raids into a manageable game of who has a better plan of beating/omitting the enemy resistance.

If both players sink massive sums of money in defenses... yeah, that's going to create an impasse alright.  Generally speaking, this is only an issue with the AI and beginner players.  Experienced players know that the starbase is a relatively poor investment since it only protects a single gravity well.  Pumping that money into frigate-power has much better returns and can usually savage an opponent who invests heavily in static defenses.  At any rate, I don't think it's fair to complain about the game stagnating when your chosen strategy is to hunker down and build defenses.

If it's a TEC or Advent starbase, just keep your distance.  The AI seldom uses these starbases to any effect, so they can largely be ignored.  Destroy the rest of the defenders away from their precious super-base, then whittle it away with bombers.  The Vasari starbase (or heavily upgraded starbases) can be a little more tricky since it will chase after you.  You can always just swarm it with bombers, but as others have pointed out the Advent have a great defensive fleet combo with the Progenitor Mothership and Iconus Guardian.  This will absorb the starbase's damage quite nicely and enable you to tank through it.  This will also work against hte TEC and Advent varieties, but both these starbases have special abilities that can devastate a fleet that gets in range, so this is not advisable.

The strength of pirates were balanced against veteran players... who generally eat unfair difficulty for breakfast.  This has made pirates totally overwhelming against lower difficulty AI's and new players.  I recommend turning the feature off while you learn.

Rarely play Sins at the moment, so my two cents for about what it's worth. The real game for me is still Vanilla Sins. Has more of the StarCrafty feel about it. Or maybe just impatient and lazy I am.

Star bases slow down the game. Mines are tedious and annoying; get nothing out of them. Diplomacy to me is more tedium on the road to nowhere. Pirates suppose to be easy trash mobs to level with.

Based on your post, where you were may be where you suppose to be. But max the fun and see where the answer is for you.

The slow-down isn't that noticable in multiplayer, where human opponents will simply ignore starbases and attack other targets.  Things are already faster paced than most people realize, so stopping to starbase all your front-line worlds really isn't viable.

To give you an idea of how starbases are used in Entrenchment, I'll describe a match I had some time ago with DirtySanchez (that one goes down in my all-time favourites, despite having an anticlamtic ending).  The map was a premade called "Asteroid Belt".  It was a 2v2, each team had an Advent and a Vasari.  I was Vasari, DirtySanchez was the other team's Advent.

Now, each player had one choke point (a volcanic world) that lead away from their local starting area and into a large web of asteroids.  I immediately sent my Jarrasul deep into the asteroid field to capture the asteroids adjacent to both the enemy volcanic worlds, thereby cutting them off from accessing the rest of the map.  I got some migrators in range and put up starbases.  Both the opponents had opened by colonizing their local area and ignoring the more distant center of the map, and weren't able to stop me from getting these bases operational.  The enemy Vasari sent his own Jarrasul running past my starbase line and colonized an asteroid in the center of the map, setting up a staging base, while DirtySanchez committed himself to building a massive fleet of illuminators (this was 1.181, when the unit was bugged and dealt boatloads of extra damage) and punching through. 

With the enemy Vasari thwarting our attempts to colonize the center of the map and drawing a substantial amount of our fleet power to contain, DirtySanchez went on the offensive and attacked one of my two super-starbases.  Repair platforms were not yet operational, so he managed to destroy the base, but he took sufficient casualties that we were able to drive him off and maintain control of the planet.  We continued to attempt to lock down the rear-facing Vasari while building up our economy.

With our economies finally beginning to pull ahead by virtue of our extra real-estate, the enemies went on an all-out rampage.  The Vasari made an offensive-starbase gambit against my Advent, while DirtySanchez bypassed my starbases and made a bee-line for my capital.  Seeing things deteriorating, I starbased my rear choke points and sent my own fleet to attack the enemy's core worlds.  In the meantime, I used my Vasari advantage to research cheap emergency facilities to delay the enemy bombardment.  My goal was to capture their worlds faster than they could raze mine, and when the dust settled I intended to control all the choke points with my starbases.  At this point, my ally declared he was bored and left.  We called the game, since the Vasari enemy had a starbase on one of the Advent's worlds, which would cause the AI to continually suicide its fleet.

As you can see, this was an extremely starbase heavy game.  I had maybe 5 or 6 heavily upgraded ones at the end, but the enemy remained actively on the offensive the whole time and keeping us on our toes.  My starbases also kept my fleet small, forcing me to play very cautiously and giving the enemy the opportunity for their aggressive ploys.  Good players don't let starbases get in the way of their conquests, and they've added a lot to the game.

It's a joke, but not a bad joke.  The content did add to the game, just not a whole lot.

Pirates weren't balanced for veteran players, they were 'balanced' to outperform everything in the game.  E.g. instead of that epic game containing each other with starbases you could have just paid a pirate mission to smash through their home world while you make a simulataneous assault on their other resource-heavy worlds.  Pirates pose more challenge than Cruel AI opponents, who are outperforming players by a 4:1 ratio when it comes to resources.  Really only Advent late game fleets can stand up to pirates.  Anything else just becomes a suicidal waste of fleet resources.  They also favor the TEC early game when teams are involved, as Worthy Cause gives the TEC player tons of extra cash to pay on bounty and missions.

Mines are GOOD!!!!!!!!!!

seriously... it's really annoying when someone uses it.

Use scouts to detect them, then you can destroy them. A few scouts and a carrier can make short work of them.

hit them with anti structure cruisers and bomber, but try to stay at a distance, in some cases, going around the planet is a good idea if your looking to bomb, however if they have the research your best bet is to kill everything and then gank the starbase last.

i might try the flak+scout thing that seems like it would work best

to me the pirates need nerfed just a little bit or at least the raids should be as large as the bounty not for 100 credits be this insane fleet that are planet desolators.