Does paying child support give you more rights than the court has already given you?
The situation is this, the father of the child just 2 months out of prison, he is a career criminal and drug abuser, was given joint custody by the court because the person from family services did not give her the option to see the judge, the mother was told that joint custody would be granted regardless of what the mother did, that we just found out is false, but now the mother want child support from the father but does not want to give him any more time with the child, he gets 2 phone calls a week and 2 supervised visits a month at a supervised visitation center.

No, child support has no bearing on visitation... they are legally 2 separate issues.

What I'd like to know is how mother is going to have father pay ANY child support if he is a career criminal as you state and is in and out of prison?

No, paying child support does not give him any more rights to more phone calls or visits other than what the judge originally ordered unless the mother decides to give father any more and does this on her own (which is not a good idea by the way). What the judge orders stands as a guideline for the parties to follow. If the father wants this arrangement changed and the mother does not agree to it, then the Judge will have to make another ruling on just how much visitation is increased or not increased. The father would have to petition the court for increased visitation rights.

Paying child support does not give any extra rights

And not paying does not take away any rights.
All rights are spelled out in the child custody agreement as to visitation, custody agreements and more. If you need it modified, you file a motion with the court to change the terms of the visits, custoday agreement child support and more.