There can be many causes for a burned shore power cord but sometimes a simple preventative step can save your boat.  If you are sure that the cord is not being overloaded, a common cause for a burned power cord can be a poor mechanical connection to the power box.  To help prevent this you can support the cord with a length of line tied to the shore power pedestal which can relieve some of the strain on the connector of the cord!

photo credit - http://www.qualitymarineservices.net/safety.