After doing some more research on this question I have determined, like the other answers, the reason that the miners do not boil to death is that they cool the mine actively. Originally this was done by using chilled service water and ice slurries, a method pioneered scientifically by Wagner (Wagner, H. The management of heat flow in deep mines. Geomechan. Tunn. 2011, 4, 157–163). In more recent years the depth of the mine has become such that it is economic to use hard ice. This is done by conveying hard ice into the bottom of the mine and pumping out the melt water. Here is a picture of the ice as it enters the mine:
The amount of ice needed is computed in the following way:
After the ice melts, the melt water at temperature 25 °C is pumped back to the surface. According to Howden, the company that provides the technology to cool the mine, the virgin rock temperature at the bottom of the mine is 55 °C and the ambient air must be maintained at a temperature of 28 °C or lower. It is important to realize that virgin rock temperatures are only one source of heat. Mining machinery engines, lighting systems and auto-compression of air all add more heat. Auto compression is the term used to describe the way air is compressed by its own weight as it descends into the mine. The heat flow into the mine is mitigated by using thick concrete-based insulation that provides a thermal barrier to the virgin rock. The mine is lined with this insulation, reducing the amount of cooling ice needed.
Low Geothermal Gradient
The reason why the virgin rock temperature of the mine at the deepest levels is only about 55 °C as opposed to the 125 °C we might expect is that the geothermal gradient can differ substantially depending on the kind of rocks in the local area. Different rocks have different heat capacities and insulating qualities, therefore, the thermal flux at different depths and geographic locations will be quite different. Also, tectonic effects like volcanism can increase the heat flux. In addition to this, some rocks, like granites, actually generate heat, so that can affect the heat flux, as well. Yet another factor is the orientation of the country rock. For example, quartz is highly anisotropic so the heat flow through it depends significantly on the direction the quartz crystals are oriented.
In the Witwatersrand Basin, where the Mponeng Mine is located the geothermal gradient is noticeably low and is nominally rated at 9 °C/km. This yields an expected temperature of 4 km × 9 °C/km + 25 °C = 61 °C which is quite near the measured maximums of approximately 55 °C. The apparent reason for this coolness is the presence of a thick, cratonic lithosphere in this region (Jones, M. Q. W. "Heat flow in the Witwatersrand Basin and environs and its significance for the South African shield geotherm and lithosphere thickness." Journal of Geophysical Research: Solid Earth 93.B4 (1988): 3243–3260.) This particular craton is known as the Kaapvaal Craton. The craton shields and insulates the surface layers of the earth from the hot interior of the earth. The map below shows the location of the mine with respect to the Kaapvaal Craton:
The mine is located at the white dot. As you can see it is located right above the center of the craton.