I assume that one of these serial inputs is on the hardware serial pin 0. Presumably that's the boot loader getting in on the act, thinking that a sketch is on the way. AFAIK you can't stop it, maybe you'll have to flash your program into a clean chip.
@Bob:- that link is the auto reset on the main serial connection. This seems to be a problem with two of the extra serial ports on a mega, not the main one which is still called serial (not serial1).
Aah the MEGA, sorry didn't spot that. In that case there's no reason for this to happen. Is there a shared interrupt somewhere and the Mega boot loader was never tested with incoming serial on the other ports?