[The Congress shall have Power . . . ] To coin Money, regulate the Value thereof, and of foreign Coin, and fix the Standard of Weights and Measures; . . .
Article I Legislative Branch
Section 10 Powers Denied States
Clause 1 Proscribed Powers
No State shall enter into any Treaty, Alliance, or Confederation; grant Letters of Marque and Reprisal; coin Money; emit Bills of Credit; make any Thing but gold and silver Coin a Tender in Payment of Debts; pass any Bill of Attainder, ex post facto Law, or Law impairing the Obligation of Contracts, or grant any Title of Nobility.
The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.
And this seems relevant...
Which part gives them power to ban alternative currencies?
This?
No state has to recognize an unconstitutional currency.