Can I prevent bad-paying customers from enrolling?

Yes. Set a bad-paying customer as a Problem Account by going to the Summary tab on their Family page and checking the Problem Account checkbox and save changes.  Next, go to Tools > Parent Portal > Settings and in the Class Enrollment Settings area check Prevent Problem Families from enrolling and save. 

The customer will continue to have access to the Parent Portal but will not be able to enroll.  To deny their access to the Parent Portal, open their Contact record and in the Portal Information area deselect the "Allow to Login to Parent Portal' field.