Matt K:
The Red wire is the large wire from the Starter Solenoid, used when starting the Motor.
It is large in size to handle the higher amperage.
There could be a number of reasons for the condition you describe. Some could be: (A) Bad terminal connection, (B) Loose terminal connection, (C) Cracked or broken terminal insulator, (D) Bad cable connector, (E) Bad cable (worn or high resistance), and (F) internal problems within the S/G.
I would suggest a real close look at this connection. This may require taking the S/G to a repair shop for indepth examination if this is beyond your level of experience.