Influence is as much of a defense and a weapon as your fleets are now.  Some ways around it-

1) look for the enemy planet with the most influence and conquer that one first. Let it's influence flip the others. Only the capitals are normally immune to influence

2) Use your fleet to clear out their starbases, asteroids and shipyards and build an influence base right next to a vulnerable planet to flip it.

3) get the new artifact that gives 10K influence to a planet, and colonize an otherwise worthless planet and make it the "15 minute wonder". 10K influence has about a 30 hex radius of effect.