The State is always involved in criminal cases because criminal cases by definition involve violations of laws passed by the government, whether state or federal. Law enforcement agencies like state and city police departments and the Federal Bureau of Investigation at the federal level enforce the laws. Prosecutors, then, decide whether there is enough evidence indicating a particular individual or group of people are guilty of violating the law to make it worth their while to take the case to court. Consequently, all criminal cases have the government an active participant.
Civil cases are vastly different than criminal ones, and might not involve a violation of the law. Civil cases might simply involve one neighbor suing another neighbor in court over some minor difference over where one's property ends and the other's begins. In such cases, there is no prosecutor and, hence, no role for the government other than as adjudicator, in other words, the judge. And not all civil cases are necessarily resolved in a court. The parties to the dispute might decide it is less expensive to use an arbitrator to decide which side is in the right.
The government does, on occasion, involve itself in civil cases. With the advent of "hate crimes," for example, the government might file suit against the perpetrator for violating another person's civil rights. The federal government has filed such suits against racists on behalf of the victims of racially-motivated hostile acts.