Why does my stepper motor run when I open the Serial port?
When you open or reopen the Serial port, the Arduino resets (depending on how you open it; some programs don't send the signal on DTR or RTS, which result in the reset).
My guess is, that the stepper driver reacts to the Arduino configuring its pins for the stepper. Difficult to say, what exactly happens there. To mitigate that, you can add an external pulldown resistor (assuming you have an active high stepper driver. For active low use a pullup resistor). That should keep the step line of the driver low even when its not actively driven by the Arduino.