If the country were to "go off the fiscal cliff" there would be a lack of funding for foreign aid to our allies that support are interests globally. There would be a dramatic lack of funding for the military. This would also ultimately lead to a weaker foreign policy agenda. Given the current debate it appears that catastrophe will be diverted at least temporarily.
It is true that the Defense Authorization has been passed. However, it is my understanding that a deal still needs to be reached after the extension of funding expires to ensure that foreign policy is not affected. Even if a deal is reached, foreign aid may still be impacted and that can have some unintended consequences in unsettled areas where the U.S. does not have a strong military presence. While the Defense Authorization Act will undo the original sequester, part of the Fiscal Cliff is the debt ceiling and if that issue is not resolved, even authorized funding may not be able to be paid out.
I don't see why it would. The "fiscal cliff" simply means Congress fails to enact new tax legislation after the Bush tax cuts expire. It may increase the deficit slightly, but I don't see how it would affect foreign policy by much. Foreign powers continue to buy subsidies from the Fed to this day.
If the government goes over the "cliff", then $9 billion in cuts will take place. To explain further, the deficit will go down by $487 billion because of $9 billion (1.8 percent) in spending cuts and $478 billion in increased tax revenues. These numbers show the fiscal cliff is exaggerated, and the effect on foreign policy will be negligible.