Only the legislative branch of the United States government has the power to coin money. This branch is known as Congress, and is made up of two parts: the Senate and the House of Representatives.
In Article 1, Section 8, Clause 5 of the U.S. Constitution, it specifically confers on Congress the express powers to coin money and regulate the value of it. Section 10 further states that the states do not have the power to coin money.
As the preservers of the American currency, the members of Congress are also the final arbiters of which denominations are in production. This means that not even the president gets to decide, for example, if a certain denomination of coin should be done away with because hardly anyone uses it anymore. Only Congress can decide to take a denomination out of circulation.
Article 1, Section 8, Clause 6 of the Constitution allows for Congress to punish who tries to counterfeit "the securities and current coin of the United States." It is a federal crime to mint coins or monetary systems, as some have attempted to do over the years; if found guilty, a criminal may face up to several years in prison.