The Vasari bomber is really only an issue for capital ships and starbases...like anything in SoaSE, they only become a real problem if they are spammed....and spamming can always be stopped...
Lets look at the disadvantage of bombers:
1) Light carriers require antimatter, which can be stolen by disciples, depleted by EMP, and depleted by lots of phase jumping...
2) Strikecraft can't be used in Plasma Storms
3) Bombers aren't nearly as effective against LRFs
4) If a light carrier is destroyed, it takes a while to build AND a while to build the SC squadrons...
Be creative and take advantage of these 4 disadvantages...
For fleet composition, if your enemy uses Bombers + Enforcers, then you go Bombers + LFs...LFs suck against heavy cruisers, but are by far the best at taking down carriers...rush the carriers, take them down, and swallow the losses of your LFs...Disciples are the best for this since they have steal antimatter...yeah, you suffered great losses, BUT it takes a LOT longer to build carriers AND build the squadrons than it does to build more frigates and cruisers...hoskikos are good for this as they can disable lots of HCs and prevent them from moving while you rush the carriers...
If your enemy goes Bombers + Assailants, then you go Anti-SC with support ships...you won't have any firepower, but your enemies bombers will fall...introduce HCs and Fighters only AFTER you have widdled down the bombers...
Whenever you find your enemy using a military strategy you don't know how to counter, find a way to force them to change tactics...the best way to do this is to force them into very difficult decisions...
If you attack, bring in siege frigates and some long range cruisers like Ogrovs AFTER your fleets are engaged...either they lose the planet and structures, or they must micro-manage the bombers to take out those ships...if you separate your fleet from these "diversions", their bombers are away from Anti-SC support AND will not be taking out your HCs...
If you have the option, force your enemy to make many phase jumps...use feints, use wormholes, use Sova's with embargo or Ogrovs or siege frigates or Advent starbases with culture facilities...anything that forces the enemy to make a phase jump or two...that kills the antimatter on those carriers...
Ultimately, if you are Advent you should NEVER be outdone by SC...a lvl 6 progenitor or 2 with resurrection will save those high level cap ships, and your Halcyons + Fighters should be able to take down Vasari bombers easily...their bombers HAVE to be over YOUR fleet...use Repulsion to keep away those Sentinels so your fighters aren't shot down...
If you are TEC, you should have set yourself up to be in an economic advantage in the early game...produce ships faster than the Vasari player, going with the above fleet compositions...do not underestimate Dunovs, Kols, or the lvl 6 ability on the Sova which allows a kodiak to be built in 7 seconds...