Position a locking latch against the display case so half of the latch lines up with the case's door while the other half lines up with the cabinet's side board. Mark the location of each latch screw hole in the door and the side board.
Drill into each marked hole using a drill bit that's slightly smaller in diameter than the size of the installation screw's diameter. Reposition the latch over the door board and the side board.
Set a 2-inch wood screw through each hole in the latch and into the holes in the door and side boards. Tighten each screw until the top of the screw head is even with the latch surface and the latch has begun to push into the wood.
Open and close the door to ensure that the latch installation is straight after repeated use. Make any minor adjustments in position that are required for a good fit. Set a padlock in the hole in the latch and secure it by closing the padlock.