In many states, they do not come arrest you for failure to pay. Some do but I think they are few. I do not know if NY is one of them.
In many states, the normal procedure is to summon the NCP to court for a hearing to show cause ( show cause for failure to comply with a court order). Then if they find you guilty they will impose whatever punishment the law allows i.e., jail time, fines, suspending or revoking licenses, ect.
If one is served and they don't show up, then they can issue a bench warrent for failure to appear. I think the bail for that is either the full amount of the child support or a portion of it.
If there is a warrent out the person should consult with a lawyer quickly to see how they can settle this matter. Chances are, the person is going to need one because I have found that CSE doesn't work well with people without one.
If they don't have a job they need to find one quickly and petition the court to modify the amount of support until they are able to get caught up and attempt to make a payment arrangement with CSE to pay the amount plus a little extra each month for the arrears.