My first couple games I played on smaller maps and sub-normal intelligence settings to try and learn how to play the game. Influence really wasn't an issue. Now that I've moved up to a gigantic map and normal intelligence, I'm actually getting a better idea of how influence works.
For instance, early game I dominated the influence ratings, but had to stop my planet grabbing because I still haven't figured out stable economics. As my nearest neighbor, Arceans, kept expanding my influence dwindled, little by little. Nice, neat and simple; easy to understand.
Realizing early on I needed to take some preventive measures, I put embassies and extra entertainment in any planet near our border. This worked for a while. Then I saw the military chart go through the roof for the Arceans and my influence really started tumbling. Fine; I started building military ships too and maxed out trade with them to stay in their good graces.
Wasn't enough, my influence bottomed out and various races (including my old buddies the Arceans) started demanding tribute. Then the influence boundary shifted to engulf 3 of my border planets. Instantly I build 4 influence starbases and maxed out the modules. That effort pushes the boundary back just enough to keep the planets in my sphere.
I take my social spending down near minimum and pump up military then set every spacedock I've got to building military ships. While doing this I'm planning an attack across the galaxy and building up ships for that as well. My influence starts creeping up, slowly reaching even with the next lower race.
The demands stop and my influence boundary goes back where it had been most of the game.
Then I built spin control.
Then I invaded across the galaxy.
Out of no where my influence is now 2nd.
I had already near maxed out the diplomacy tech tree, but nothing mattered until I had a military strength strong enough to hurt anyone thinking of invading. Balance seems to be the key.