Your legal right is to stay sober, the dad has the right to get his son's his cut, unless you discussed this matter previously. As for as your son's drunk grandmother, if this incident did not cause any harm or in danger your son's life in any way, it's not a legal matter but a moral issue you should discuss with your baby's daddy, and perhaps the grandmother too.
If you have sufficient evidence of any incidents (not just one) that causes you to believe your son's life is in danger, keep a record and report it to the police or a family social work in your county to investigate.
The case would be filed as [name of person accused of DWI] VS. the State of New York. The State would be the plaintiff, as it is in all criminal prosecutions.
The driver picked a bad state to get nailed for DWI in, one of the very few things New York got right is it's severe penalty for DWI.
Check out the link below, harsh penalties for that offense, just as it should be! The driver can pretty much count on fines, fees and surcharges totaling more than two thousand dollars, mandatory counseling, suspended driving privileges (if the driver jumps through a few hoops they might qualify for a restricted license) ... and every car they drive for at least the next year will have to have an ignition interlock, even for a first offense! The driver would be responsible for any fees or monitoring charges associated with using the interlock.
The driver should have a lawyer to represent them in court. Given the severe penalties that are on the table ... they should consider a DWI defense specialist.