Computers have been around for a long time but when did mainstream digital computers really begin to appear in TV shows or films? I believe this is inline and similar with this question.
From IMDB synopsis: "Two novice thieves are plotting to rob a bank in Vancouver. A photographer snaps a shot of one thief as he is carrying the bank building's blueprints. The would-be thief then begins a relationship with the photographer and attempts to retrieve the photos. Meanwhile, the thieves' plot consists of this: one man will enter the bank building after dark, while the other man sits in a van and uses a computer to unlock the building's doors."
The first consumer grade computer used for a character's personal endeavors as opposed to the quasi-business use above was an Apple II in Evilspeak (1981).
Large mainframe computers began showing up in films as early as the mid 1950s.
While by no means an actual personal computer, the first example of a contemporary home with a computer in a TV show was Whizzo on The Addams Family.