If the state agency for child support with jurisdiction over your support obligation reports you to be behind in payments, the US State Department will probably deny your passport application if the amount owed is more than $2,500. It used to be a $5,000 threshold so the amount could change again in the future. You should speak with a lawyer to what can be done.
If you have any questions regarding immigration, please feel free to call me, immigration attorney Shawn Mesa at (813) 679-5780.
Disclaimer: This information is provided as a public service and not intended to establish an attorney-client relationship. Any reliance on this information is taken at your own risk.