If the Secretary of State receives a certification from a state child support agency that meets the requirements of 42 USC 654 (31) and that certification indicates a person owes $ 5000 or more in child support arrears, the Secretary of State may refuse to issue him or her a passport, or may revoke, restrict or limit the passport, hence limiting his or her ability to travel outside the U.S.
Because the child support enforcement via withholding of passports was so successful, in the past year or so, the federal government lowered the threshold amount to $2500. For more information, read the this page: http://travel.state.gov/passport/ppi/family/family_863.html .
If you now have a valid passport, being in arrears of child support should not affect the passport's validity. Getting a passport the first time or renewing a passport is when the government refuses to process the passport applications of persons owing child support.
If you owe enough child support, some states may take actions against you that result in a warrant being issued for your arrest. If you have an outstanding warrant, any law enforcement officer likely can act on the warrant. If the officer does not have the authority to directly arrest you, the officer likely will call someone with the authority to come and arrest you.