Why have you got three stepper motors, it looks as if there is only one in the diagram?
Does the system work at present? I would have expected the wire to get fankled if the wire feeder was not being moved along as the motor turned.
The manual for the controller is here;
https://www.geckodrive.com/amfile/file/download/file_id/22/product_id/14/
It looks like you need a male 25DB plug to plug into the driver. then the manual says;
"Your step pulse width must be at least 2uS wide and the pulse polarity for the step and direction signals are ACTIVE HIGH".
So you write your own code to set the direction and send output pulses.
Start with just doing a fixed number of turns. Then you could maybe host a small webpage on the Arduino that you could browse to and use as an interface to set a variable number of turns - just a thought, it would mean you could control from a PC without having any code on the PC just a browser.