Approaching this from a real-life perspective, one of the earliest crime databases was that of the NCIC (National Crime Information Center) which was set up in 1967.
On January 27, 1967, the system was launched on 15 state and city computers that were tied into the FBI's central computer in Washington, D.C.—which at that time contained five files and 356,784 records on things like stolen autos, stolen license plates, stolen/missing guns, and wanted persons/fugitives. In its first year of operation, NCIC processed approximately 2.4 million transactions, an average of 5,479 transactions daily.
The first hit came in May 1967, when a New York City police officer—suspicious of a parked car—radioed in a request for an NCIC search of the license plate. Within 90 seconds, he was informed that the car had been stolen a month earlier in Boston. We got a report that the patrolman exclaimed, "It works! It works!"
A police drama named Adam-12 which ran between 1968 and 1975 appears to have been the first to incorporate NCIC checks in its plot. IMDb's blurb for the eighteenth episode of the first season, Log 112: You Blew It (broadcast on 8 Feb. 1969) reads:
Malloy and Reed conduct a traffic stop, but decide to let the man off with a warning before the NCIC check is completed. In their haste, they let a man wanted on armed robbery and weapons charges go free. The lieutenant calls the officers in to scold them for not going "by the book," particularly since another officer could have responded to the scene of what turned out to be a routine domestic dispute. Malloy and Reed must then put their being scolded behind them as they come up with a plan to nab the wanted criminal.