It has little to do with printing and more to do with confidence. No fiat is really backed by anything so it all comes down to confidence. Since most currencies are at least partially "backed" by the USD, it gains value through confidence in times of crisis regardless of how much the Fed prints as long as it seems generally within reason (to bankers, not the general public).

In reality, the whole world subsidizes the actions of the fed because everything is based on/backed by the USD and the fed's actions create liquidity that helps every country maintain their own house of cards.