Identify the correct replacement fan for your cooler. Check the manufacturer's instructions for details. Order or purchase a replacement fan that will fit your cooler. If you cannot find the model information you can remove the broken fan and take it to the parts store to match against a replacement.
Unplug the wine cooler and turn it so you have access to the back panel. The fan is on the back of the cooler. You may need to remove a screen or cover over the fan. Carefully remove the screws holding the cover to the cooler. Behind one of the covers you will find a fan. Remove the mounting screws holding the fan in place.
Pull the fan part-way out of the cooler. It's still attached to the wiring, so don't pull too hard. Look for the plug attachment for the fan by tracing the wiring back from the fan. There will be a plastic plug interfacing two sets of wires. Press the clips (if any) while pulling the plugs apart. Grip the plastic, not the wires and use firm, even pressure. Don't yank or force the plugs apart. If they won't come apart, look for clips or catches that might need to be released, release them then try again. Be patient and unplug the fan.
Plug the new fan into the wiring harness. Plug in the cooler to test the fan's operation. If it comes on normally then you can unplug the cooler again and proceed. If the fan does not start you have an electrical problem. Contact the manufacturer of your cooler for warranty and repair information.
Re-mount the fan in place with the mounting screws. Tighten them firmly, but not over-tight. Use the same amount of force as you would opening a doorknob. Replace the cover (if any was removed) and screw it back in place as well. Plug in the cooler and slide it back in place. Make sure you have at least 2 to 4 inches behind the cooler to allow for air circulation.