State laws differ. If your state law allows for a person to be taken into custody due to driving without a valid license, either a suspended license or revoked license then one could be arrested for driving with a suspended license - it would not usually matter the reason for the suspension.
In many states the driver will not be arrested on the spot but will be cited and they will not be allowed to drive the car away from the scene of the citation. So the car may be impounded, towed to the owner's home or sometimes it is allowed to be driven home by a licensed driver if the police use their discretion and permit this.
If a person is cited and convicted of driving with a suspended license and your state assesses points for this violation of the law then points would be placed on your driving record. Insurance companies' rating systems differ however driving with a suspended license will normally be a serious offense that they consider, also some companies may drop your policy altogether once they find out that you have a suspended license and not a valid license to drive on.
An indefinite suspension means that there is not a certain amount of time that it will last. So if you have an indefinite driver's license suspension due to unpaid child support then normally the suspension will last until the arrearage has been made up or at least a payment plan has been decided upon. See the examples below for what some states' do in regards to driving license suspension and delinquent child support payments.
Indiana states that if a court finds that a person is delinquent as a result of an intentional violation of an order for support, the court shall issue an order to the bureau of motor vehicles (BMV). The court order can state that the person’s driving privileges are suspended until further order of the court.
In South Carolina if the non-custodial parent is in the arrears by at least $500 and has not made a payment within 60 days, any driver’s, occupational, professional, business or commercial license issued by the State of SC is subject to being suspended or revoked. Once a license has been suspended or revoked the parent cannot have the license reinstated until they have reached an agreement for paying the arrearage.
To find out specifics about your state's laws regarding driving with a suspended license due to delinquent child support payments and how long the suspension will last, contact your local Department of Motor Vehicles.