In addition to personal reputation with you, each contact has an influence statistic, which is global and represents how influential they are to the galaxy. Normally this is between 0-100 but can be as low as -10 and at least as high as 110. Influence is required to perform some operations, such as giving a high level crew recruit or a high level trade permit.
To influence faction conflicts, take missions that directly affect the conflict - these are usually in the middle of the screen of the 3 types of mission offered by a contact and will clearly say which conflict they will affect. Doing these missions will aid that faction in said conflict.
You can also influence faction conflicts by spying and blockading - say Cadar are in a conflict with Thulun and you want to help Cadar. Find a Thulun world and blockade/spy - some of the reward cards will be "Strike a Blow in the Conflict", which will add directly to the Cadar score for that conflict. You can also sell any spying intel back to your Cadar contacts which will raise your rep and their influence.
For rumors you can potentially help out any contacts suffering influence drop due to rumors by taking an appropriate action. If you read the rumor details it will generally tell you what action can be taken to help offset it - ie you can try to run a blockade with goods for the blockaded world.
Another great source of influence is selling Rare Trade Goods - all contacts from the quadrant where you bought the goods will gain influence when you sell them. This can be fairly situational though and won't help if you don't have the right contacts to take advantage.