Is it better to detain, or reject?
Solution 1:
I'll consider it as the game is in end-game state, with the blue stamp and distant guards.
It takes about 10 seconds after pressing the detain button for the guards to have removed the person from your booth and you being able to press the button for the next person.
This means that the income from detaining equals being able to handle a person in 20 seconds from speaker to speaker, or equivalent to processing 18 persons/day.
Conclusion: Detain, not Reject.