1. Exping in beastmen claimed areas /with Signet to raise Bastok influence. You can get double points by exping with an All same nation party.
2. Expeditionary Force runs to influence zones under beastmen control
3. Garrison to retain influence in already held zones
4. Trade junk armour/weapons to the Outpost vendors to increase influence, I used to this when levelling crafts with all the crap I was churning out.

If you're not actively trying to help Bastok get first place, don't complain that the system is broken.