Given what is happening in Syria, this is a timely question. I think that one factor that does influence military intervention in another nation's politics would be humanitarian concerns. For example, the use of chemical weapons in Syria that resulted in the deaths of 1500 people, over 400 of whom were children, represents a rationale behind the United States considering military intervention in the form of action against the government of Syria. Nations that establish a "red line" point such as the use of chemical weapons and biological warfare find themselves having to contemplate intervention when that line is crossed.
The cause of humanitarianism is evident when chemical weapons are used or when genocide is happening on a large level, such as in Kosovo in the late 1990s, are examples of factors that influence the likelihood of military intervention. History does not look too kindly on Western nations that stood by while the slaughter in Rwanda raged on without any type of military intervention. Since then, nations have been mindful of the humanitarian factor when weighing intervention options around the world.