You can follow your progress toward this achivement by looking into the game's save files.

You can find them on your C drive, this is the path on my computer
C:\Users\USER NAME\AppData\LocalLow\Pixel Crow\Beat Cop\BeatCopSaves18.

Note: You may need to change your folder settings as the AppData folder is normally hidden.

After each day you can check how many thieves spawned that day and if you missed any and also check the total number of arrests. Pictured are my days 18 and 19. I was going into day 20 having arrested 1 thief and a couple mobsters extorting money and someone else, a wall tagger I think, bringing the total number of arrests to 49.

On day 20 the achievement popped right after I arrested another thief.

Example:
"thievesSpawned":2,
"achv_arrest":45,