How can I repair the bottom fan of my Kenmore Elite side-by-side refrigerator? It is no longer turning.

Test it to make sure it is/is not working by bypassing the circuit and going directly to the fan. If the fan works, it is something else. If it does not work, it is most likely bad. Could be the compressor snap switch if so equipped.

Here is a manual: https://dl.dropboxusercontent.com/u/8010163/Whirlpool%20Side%20x%20Side%20Refrigerators%20and% 20Freezers%20Service%20Manual.pdf
I do not know about your particular unit, but some of the new refrigerators with electronic controls use a 12 volt fan, so check it with a digital ohm meter by reading between the two power terminals on the motor to see if you have continuity. Do not run the refrigerator without this fan running or it will burn out the compressor. Also, one you fix it make sure you clean the coils with spray soap and spray it off with a little bit of water from a hose nozzle and vacuum it up with a wet-vac. The fan motor is a special type and you will find it here: http://www.repairclinic.com/Shop-For-Parts/a4b6c53/Kenmore-Refrigerator-Motor-Parts or https://www.midwestapplianceparts.com/index.php?main_page=index&cPath=65_169_946&zenid=gio7rfqfvpkva8abrbfocfcu37
Do not just replace it unless it stuck or has an open winding because there is a control for that fan motor that can be bad, too.