Wayne Gray and John Scholz have published research indicating that OSHA inspections do make a difference. You can check out their paper "Do OSHA Inspections Reduce Injuries? A Panel Analysis." Also "Inside the Black Box," co-authored by Gray and John Mendeloff, which was published in the March 2005 issue of Law and Policy.The first paper they mentioned, "Do OSHA Inspections Reduce Injuries? A Panel Analysis," was published in 1991. Gray and Mendeloff followed that up with the paper, "The Declining Effects of OSHA Inspections on Manufacturing Injuries: 1979 to 1998," published in 2002, which shows a decline in OSHA effectiveness. I can't access the second reference, "Inside the Black Box: How do OSHA Inspections Lead to Reductions in Workplace Injuries?" so I'm not sure what it says.
It's interesting that these papers come from the National Bureau of Economic Research, not from the S&H field, although at least some of the funding for the research was provided by NIOSH.