First post, by gumpy
F1 and on should work fine?
Mostly the F1..F15 keys are captured by the host system.
Probably press another key along with the F1 key in order to get recognized by the DOSBox game (the Fn key), which often works in fullscreen only.
The game has an issue with extended keys that might also happen on real systems with enhanced keyboards. DOSBox simulates a standard keyboard to some extent with CGA, Tandy, and Hercules machine types, but there are still some enhanced keyboard behaviors in the emulated hardware and BIOS.
The issue appears when you press an extended key, such as a cursor key, and subsequently press the left Ctrl or Alt key, which causes the system to think the right Ctrl or Alt key is held down. When those shift keys appear to be held down the function keys will not be recognized because they are shifted. It's actually quite easy to trigger the condition because of changing cycles with Ctrl-F11/F12.
You can avoid the condition by using the keypad instead of the cursor keys, or the right Ctrl or Alt instead of the left, but if you do accidentally trigger the condition it can be cleared by pressing and releasing the right Ctrl or Alt key. For extra convenience I've attached a small program that prevents the issue; put it in the game folder and run it in DOSBox to start the game.
Airborne Ranger has a selection of video types to choose from at startup. There is an MCGA choice that uses mode 13h, but there is no mention of VGA. It's not conclusive evidence, but considering the 1988 release date it suggests to me that the game would have been tried on a PS/2 (model 30 at least) during its development.
Another plausible explanation for why the problem escaped notice is that there seems to be no compelling reason to use the Ctrl or Alt keys while the game is running on a real DOS system.