I remember reading about producer influence in regards to snatching idols from other groups a while ago. If I remember correctly, among other factors, the higher your producer's influence level, the higher the chance of you successfully convincing an idol from another group to join your own. Keep in mind that this has not been implemented into the game yet, and was/is part of the planned asynchronous multiplayer features. Not much has been said about such features since the end of the Kickstarter campaign, however, so this may be subject to change.
Other than that, all I can see it doing is maybe how easy it is to convince idols to postpone their graduations and similar scenarios, regardless of how much personal influence you may have over than individual person.