Title says it all

First, and most obviously, are the technologies and system improvements that lower disapproval.  Beyond that, the strategy for handing approval depends on your faction and policies.  I'm running a Lumeris game right now that is curb-stomping the approval barrier.  They start optimistic, which is nice, but I've also assimilated the Z'vali and Kalgeros species, which have great empire wide traits for approval.  There are also some really good Pacifist policies for approval benefits, such as the Double Bliss Bill that doubles the effects of approval (or disapproval) on a system, and the Love Not War Rule that gives flat approval bumps for being at peace or allied.  As far as I've noticed all the policy types have some way of affecting your approval, which means putting the party you want in power is important too.  The Militarists have one that gives you +25 just for having a fleet in orbit (any cheap ship will do). It all ties so neatly and satisfyingly together.  Heros and their skills can play an important part in approval management too.