What rights do fathers have to see their children when the mother is claiming they are to young to be without them and father got a paternity test within a week of baby being born and now is being denied the right to visit with his son without the mother being right there?

He has the right to go to family court and petition the court for joint custody and visitation rights. Then if she does not follow through with the orders he can go back to court to get the Judge to reinforce it.
Other than going to court he is at the mercy of when she says he can see the baby.

He has every right that the family court will give them.

Was the DNA test though the courts. If not you need to go to court and get a custody agreement and child support thought the courts.

At times some courts give less visits for the first 6 months or require parenting classes, but you get a court order and then go to court to keep it inforced