In the past the technician has received many complaints about rudeness regarding this coworker How should the technician handle this complaint?

Politely. Most people tend to forget that without customers there would be no company, no phone to answer, and no technician. Without knowing the circumstances, I would suggest that the technician communicate the customer's complaint to the co-worker so that an appropriate response can be formulated. These days (and in this competitive environment), the customer can (and will) go elsewhere for any number of reasons - price being the most popular. You have to do it better (and cheaper) than your competition.