A reserve currency is a foreign currency that is traditionally held in countries’ official reserves because of its global importance as a medium of exchange and its inherent stability.
The reserve-currency country enjoys the consumption benefit of running a trade deficit, while the rest of the world benefits from the additional liquidity, which helps facilitate trade.
The cost comes from the declining value and credibility of any currency which runs a persistent trade deficit - eventually leading to a reluctance of creditors to hold the reserve currency. 
As Francis Warnock (professor at the University of Virginia's Darden School of Business) points out in a paper for the Council On Foreign Relations, in 2010, the US confronted a dilemma first identified in 1960 by the Belgian-born Yale economist Robert Triffin.
To supply the world’s risk-free asset, the country at the heart of the international monetary system has to run a current account deficit. In doing so, it becomes more indebted to foreigners until the risk-free asset ceases to be risk-free.