Can father sue for visitation rights if he doesn't pay any child support?

An obligation to support a child is independent of the rights of the parent and child to be with each other, i.e., you cannot condition rights of visitation with an obligation to pay child support.  Have you done anything to set and/or enforce the father's obligation to support the child?  If not, you should do so.  Access to a child is usually granted unless there is a good reason why it would not be in the best interests of the child.  Has the father acted appropriately or inappropritately around you and/or the child?  Has he done anything which raises his fittness to see the child?  These questions should be answered appropriately if it is your desire to limit to prevent contact.

It will not look good for the father if he does not support his children.  Although as pointed out above, the obligation to pay child support is independent to visitation.  You should file a child support petition so that you are entitled to receive it.  You can not get retroactive support until you file unless someone is on public assistance.  In that event a child support order can go retroactive but some money would be used to pay back public assistance.  I am a New York family law attorney.  Give me a call if you would like to discuss the matter.  www.NYAttorneyDivorce.com www.CriminalDefense1.com

As a general rule, failure to pay child support (especially if it is not in violation of a court order) will not result in a father's termination of his parental rights.  Thus, it is entirely possible that a father who is not ordered to pay child support to have/retain visitation/custody rights for his children.  Of course, willingness and/or refusal to pay child support may factor into a judge's decision when making a visitation order.  In any event, in response to your specific question, what you need to do is to hire an attorney who is experienced in family law to represent you before the court.  Good luck.