N.Y. Uniform Dist. Ct. Act § 2408
The judges of the court shall have access to and possession of the court-rooms and court offices and other places provided by the first judicial district for the transaction of the business of the district court. It shall be an obligation of the first judicial district to supply and pay for whatever may be necessary for the transaction of the business of said court and the judges thereof, and to supply all proper accommodations, books, stationery and furniture and to pay all salaries, compensations and expenses and disbursements herein authorized; and the proper county authorities shall annually include in the county budget chargeable only to the first judicial district such sums as may be necessary to pay the same. Only the members of the county board of supervisors from the towns comprising the first judicial district shall be eligible to vote upon the district court budget. All fees and other revenue of the district court system shall be credited to the first judicial district.
N.Y. Uniform District Court § 2408