It's easy to measure how safe you are in terms of the number of injuries or accidents on a work site or within an office, but the number that is reported doesn't tell the full story. If you have a number of close calls, then your luck won't last forever and it's likely indicative that you policies need to be changed.
Encouraging Communication
You can't be everywhere at all times, which is why you'll need people to be entirely honest with you about what they've been doing on the job. Employees might assume something was an isolated incident or just a mistake on their end, and those kinds of assumptions could spell trouble for your company. You can't prevent every incident, but you can start to see patterns emerge when you make it clear that information is valued when it comes to their day-to-day activities. By definition of almost, nothing has happened. It's easy for us to forget about things that didn't happen which is another reason why it's often not reported. So instead of being infuriated by having a team meeting where one person brings it up and then 6 other people agree, work out a way to talk to your people or have an official reporting system about would-be accidents.
Proper Follow-Up
It's quite possible that workers are requesting changes that aren't possible on a short-term basis or ones that are too costly for the company to consider right now. You'll need to think of the proper work-around solution that can minimize the chances of something else happening. This can be as simple as having one team member assist another during difficult tasks as an extra pair of hands may do the job. There also is something to be said for having your best people on the job, as some can utilize their skill sets more. If you do have an employee who can consistently perform a difficult task whereas others seem to think they may be hurt at some point, then you should find how to use everyone's strengths to the company's advantage.