There are many ways mirrors can be protected.
A space most at risk for breakages would be in the free weights area in the gymnasium. The most affordable protection is an Aluminium checker plate adhered to the required area with mastic.

Blackout curtains offer a certain amount of protection in multi-use areas, for example basketball courts require extra protection, this can be obtained by impact pads specially manufactured to size using high density foam and hung from high level fixings. These can be installed in minutes and save a fortune in mirror repairs