FriendJosh:
Destabilization is one of the few (in fact, only) reasons I build News Networks. Their 10% "resistance", apparently, is effective against this effort. (I say apparently because I have no empirical data/datum to base this on. Instead, I have merely observed that, in Large/Huge games where the Torians pursue an influence strategy, News Networks on my border worlds appear to retard their efforts to culture flip me.)
Destabilizing an opponent will accomplish two things: (a) the AI empire will lower its spending (and perhaps its taxes) to raise its overall morale; and (b) the I-League (if already present) will gain any planets the AI allows to go into revolt.
In this way, you can use destabilization to "soften up" an opponent. Frequently, should I border 3 AI's (very common on a Huge map), I will cooperate ("ally") with one, declare war on the second, and destabilize the third. This allows me to deploy my Fleet to a flank without worrying about invasion from the opposite end(s).
~SDC~