Aside from biological functions like breathing, the best use of air is for energy. In order to burn anything, it must react with oxygen. You can credit the energy produced by fossil fuels as much to oxygen in the air as you could to the fuel itself!
You might argue that air is a resource in pneumatic pressure systems as well, using the air as a medium for storing energy as pressure.
Air is also “mined” for trace elements through air separation. For the most part, it is nitrogen that is extracted, because it accounts for 78% of gases in the atmosphere. Trace elements like noble gases, such as argon, are also extracted in this way.