1st step: watch your factions. By doing what they want (open borders/closed borders, etc.) they'll give more influence.
2nd step: rivalries. It's weird, but the more rivalries you have, the more influence you'll get.

If you're struggling for influence, you're expanding too aggressively, too soon... There's really no other use for influence in STNH short of integrating, or subjugating other empires...

An easy way to get additional Influence is to build lots of science ships and set them for auto-explore, along with stacking Envoys, so that you can work through all the First Contact events as each will award a lump sum of influence...