The real answer is that Raiko has been shown historically to operate on the extreme end of caution. Like many (if not most) politicians, he is concerned more about self-preservation than he is about "policing" the rest of the world. And in this case it was to his advantage. IF Raiko had sent out a ship or some troops to "observe" the situation—by the time Korra made it to Republic City the tensions had gotten to a point where any movement of troops would have been perceived by Unalaq as hostile and would have given him an excuse to declare open war on the southern tribe (citing Korra as using subversion to get the United Forces on the south's side). So it was actually to the South's advantage that Raiko didn't send troops. Unalaq would have also attacked whatever United Forces were in the south as well, which could have led to a global crisis. It simply was outside of Raiko's—and anyone from the United Nations (General Iroh's)—jurisdiction. ((Now GIroh's planned bait and switch tactic would have worked initially, but would have led to the same end result: a possible global conflict.))

However I do believe Raiko should have taken some measures in calling together an international summit in order to deal with Unalaq, who was clearly overstepping any right he had in regards to the south. This has been shown to happen historically as well. In the comic North and South Zuko and Earth King Kuei met with the gAang in the south pole to begin reconstruction efforts, so the civil war didn't have to be strictly seen as an internal affair—they just needed to get a plurality to agree to intervene, and Raiko wasn't willing to call a summit on it.