The Secretary of Treasury is authorized and obligated to print United States currency by statute, codified at 31 U.S. Code § 5114.
This is an exercise of the enumerated power to coin money: the "coinage clause." The Legal Tender Cases and the case that you cite (Juilliard v. Greenman, 110 U.S. 421 (1884)) are the leading authority on the scope of the coinage clause. In those cases, the Supreme Court held that the coinange clause grants to Congress the power to issue paper money that is legal tender. That power has not been challenged since.
Juilliard v. Greenman:
The states are forbidden, but Congress is expressly authorized, to coin money. The states are prohibited from emitting bills of credit, but Congress, which is neither expressly authorized nor expressly forbidden to do so, has, as we have already seen, been held to have the power of emitting bills of credit, and of making every provision for their circulation as currency short of giving them the quality of legal tender for private debts, even by those who have denied its authority to give them this quality.
It appears to us to follow as a logical and necessary consequence that Congress has the power to issue the obligations of the United States in such form, and to impress upon them such qualities as currency for the purchase of merchandise and the payment of debts, as accord with the usage of sovereign governments.
... we are irresistibly impelled to the conclusion that the impressing upon the Treasury notes of the United States the quality of being a legal tender in payment of private debts is an appropriate means, conducive and plainly adapted to the execution of the undoubted powers of Congress, consistent with the letter and spirit of the Constitution, and therefore within the meaning of that instrument ...
The reasoning in this case was not based on the paper being exchangable for metal, so any changes in that regard since will not have affected the outcome.
For a deeper history of how this power has been understood, and for an originalist lens on the topic, see Robert G. Natelson, "Paper Money and the Original Understanding of the Coinage Clause".