First, we are talking about 1.09+, not 1.05.
You pretty much need your own fighter coverage to eliminate the enemy bombers. Luckily, fighters destroy bombers, so you don't need as many, but you still have to dedicate a chunk of your fleet to carriers or you are going to have a hard time with enemies who are carrier heavy. Once the bombers are contained, your enforcers have much higher survivability to do their job. Once the bombers are dead, use your fighters to kill enemy LRM and enemy fighters.
If the enemy doesn't have LRM's (or so few you can kill them easily), Skirmishers are also a good idea.
It is pretty critical you research the Vasari self repair tech (2nd tier military) for the enforcers and skirmishers. For the cost, Vasari has less firepower than TEC, but the ships are tough, and the self repair ability is critical to keep those expensive HC's alive as long as possible.
Focus fire the enemy Hoshiko's and then go after the carriers. Upgraded Assailants are great for this, UNLESS you can't keep them safe from the enemy fighters, in which case they will die.
The battle between Vasari and TEC becomes an economic one at some point. As Vasari, you have to be able to control the neutral extractors on a map and establish yourself very quickly early game. TEC has good economics and cheap ships. If you deadlock without a major advantage, these factors really start to hurt Vasari as you pass midgame where people have econ and advanced ships out. You will need Returning Armada or Subverters at some point. RA rushing may no longer be viable, but Tier 2 RA is still a great way of getting influxes of cheap ships. If not returning Armada, you will need Subverters. Without Subverters or Returning Armada, an equally skilled TEC player will eventually get you if you haven't killed him early. If you have time to get 15-20 Overseers, you have time for Subverters, which I consider more powerful. But you still need those fighters, or the enemy bombers will maul your support cruisers.
Sorry for the long answer, but the good side of this is fleet diversity is tremendously rewarding in 1.09....as opposed to 1.05.