Don't think geothermal works very well with air directly, they normally use liquid and heat exchangers. And you might need 1000s (maybe 10s of 1000s...) of feet of liquid pipe buried pretty deep in the desert to support the # of miners you can pack into a 900 sq ft shack.

I think you'd probably be better off using an evaporative cooling system, they work really well in dry places and can be made DIY pretty cheap.