In these child support cases, generally, individuals who fail to pay child support are held in civil contempt of court. This essentially means that the individual willfully refused to comply with the court ordered child support by not paying. Since it is a willful refusal (i.e., the person could have paid but did not), the theory is that the individual holds the "jail keys" in their own pocket and can purge the contempt by making the ordered payments. The individual can be held in jail until the civil contempt is purged by payment. In reality, whether the individual can pay the child support is somewhat irrelevant as they often get arrested for failing to pay under the presumption that they could pay since the court ordered the child support and then it becomes an issue of whether it was willful under the circumstances. Often times, bail is set and the bail goes to pay the delinquent child support. You would be surprised how frequently individuals who refused to pay while out in the real world are suddenly able to find $1,500.00 to bail themselves out. Thus, the bail goes to pay some of the child support that is owed. That is how being held in jail works at times to pay the child support. It is not a perfect system but is, unfortunately, the way it frequently works. Of course, the point is well taken that holding individuals who do not have a job and do not have a way of paying in jail is counterproductive to them getting a job and paying. In those circumstances, they are frequently released without paying bond.
Sorry for the long and boring legal answer.