The appropriate remedy for someone who does not pay their child support is go in front of a judge and have the judge determine the appropriate solution.
If the court initially ordered child support to be paid, then the appropriate remedy is to file a Motion for Contempt for their failure to pay the required child support.
If the child support was an agreement between the parties and was not incorporated into a court order, then you would have to Motion the Court to require the party to pay child support.
One should not refuse visitation with the parent based on the fact that child support is not paid. If custody and support were ordered by the judge, and you refuse visitation, you may get into a situation whereby you have both violated the order of the court (one for child custody and the other for failure to pay child support).