You need a male 25DB connector.
At face value you only need four connections to that;
- Pin 2 X axis Step (output from Arduino)
- Pin 3 X axis direction (output from Arduino)
- Pin 15 Fault (input to Arduino)
- Pin 25 GND
You can use an Arduino Uno.
You need to check exactly how the I/O should be wired to the UNO electronics is not my area and I don't want you to zap anything.
You will need a USB A-B Cable to connect to a PC running the IDE for programming purposes.
https://www.robotshop.com/uk/usb-cable-a-b arduino.html?gclid=EAIaIQobChMIvOOazYCI2wIVsQrTCh3L1goUEAQYBSABEgKfM_D_BwE
You then write your own code.
As mentioned you may also want an ethernet or wireless shield so that you can make a user friendly interface to run the machine from the PC - but you don't need to do that at first.
When you run this it will still be manual because somebody will still need to feed the wire by hand unless you build some kind of mechanical following mechanism.
You will probably also want an input so that whoever is feeding the wire can get the stepper to start. A press button might be an idea so that the motor stops when the button is released in case the person feeding the wire gets tangled. A E-Stop to cut the power would also be an idea.