What is the primary responsibility of the New York State Unified Court System?

The primary responsibility of the New York State Unified Court System is to administer the state's court system and procedures. This includes managing the various courts across New York, ensuring that they function efficiently and fairly, and providing the necessary resources and support to judges, clerks, and court personnel. The Unified Court System also plays a crucial role in upholding the rule of law by handling civil and criminal cases, protecting rights, and providing access to justice for all citizens.

Each of the other options relates to established governmental functions but does not pertain to the specific duties of the court system. Administering the prison system, overseeing education policies, and regulating financial transactions are responsibilities typically assigned to other branches of state government or specific agencies designed for those purposes. Thus, option C clearly stands out as the correct choice by directly aligning with the core mission of the New York State Unified Court System.