Not commenting one way or the other, but here is what authorizes this:

The Personal Responsibility and Work Opportunity Reconciliation Act of 1996 (PRWORA)[0]

SEC. 370. DENIAL OF PASSPORTS FOR NONPAYMENT OF CHILD SUPPORT. (a) HHS Certification Procedure.— (1) Secretarial responsibility.—Section 452 (42 U.S.C. 652), as amended by section 345 of this Act, is amended by adding at the end the following new subsection: “(k)(1) If the Secretary receives a certification by a State agency in accordance with the requirements of section 454(31) that an individual owes arrearages of child support in an amount exceeding $5,000, the Secretary shall transmit such certification to the Secretary of State for action (with respect to denial, revocation, or limitation of passports) pursuant to paragraph (2). “(2) The Secretary of State shall, upon certification by the Secretary transmitted under paragraph (1), refuse to issue a passport to such individual, and may revoke, restrict, or limit a passport issued previously to such individual.[1]

The above may have predated the amended copy, as a threshold of $2,500 seemed to be the case at least 3 years ago, for whatever that is worth.

[0] https://en.wikipedia.org/wiki/Personal_Responsibility_and_Wo...

[1] https://www.govinfo.gov/content/pkg/COMPS-1793/uslm/COMPS-17...