I've thought about this some, and I think it should be possible to cool an audio gear cabinet, such as you plan, while keeping the front door closed.
The back of the cabinets should be closed and two or three fans (120 to 200 mm diameter) should be mounted in a row across the top of the rear panel. These fans should be mounted so they remove air from the inside of the cabinet and blow it outside.
If the lower edge of the front doors are about 10-12 cm above the floor, fresh air can enter through vents built beneath these doors. They can be both in the front and in the rear. These vents should be as wide as possible.
Build the cabinet shelves with many wide slots cut through them to allow easy passage of air from the vents near the floor, up through the shelves, and out the back of the cabinet through the fans.
I liked the various choices of fans and mounting hardware available here:
Need to cool off your new receiver? What about your new game console or DVR that is stuck in your A/V cabinet? We have a number of solutions to help cool things off * *** Please checkout out our FAQ information page, it talks and has additional video
The most important specification of a fan is how much air it moves. One cubic foot per minute (cfm) = 28.3 Liters per minute. Fans with variable speed will be quieter at slower speed, and move less air. Large fans, 200 mm, will move much more air at slower fan speeds and less noise than smaller 100-120 mm fans.
Of course, details matter here. The actual dimensions of your cabinets, and the volume of air inside them will dictate what size and how many fans to use. As a guess, I would plan to have fans capable of moving at least the entire volume of air inside the cabinet once every 5 to 10 minutes. More is better, within reason. Once you know what the size of your cabinet is, it will be easier to decide how many fans of what size fit across the width.
Good luck with your plans. Please let us know what happens.