Can I prevent bad-paying customers from enrolling?

Yes. To do this you have to identify a bad-paying customer as a Problem Account. Go to the Summary tab on their Family page and check the Problem Account check box and Save Changes.  

Next, go to Gear (icon) > Settings > Parent Portal > Settings (tab) and in the Class Enrollment Settings area check Prevent Problem Families from enrolling and Save Changes.

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.