When it's working right, progress is saved automatically at the start of each day when the newspaper appears. If you've played several days and no saves appear on the STORY screen, there's a problem.
Try this:
- Run PapersPlease.exe in Administrator mode
- Make sure your user directory is not symlinked (mklink /d). Junctions should work ok (mklink /j) but user directory symlinks are not supported. If your user directory is on a separate drive and symlinked to C:\Users\, the game will fail to save.