Is there some issue that I should be aware of with connecting the Arduino's ground to the welder's ground like this?
Yes. If whatever is powering the Arduino (e.g. a PC via the USB port) has ground connected to mains ground, and so does the welder, then you will have a ground loop. Ground loops are generally a bad thing, particularly when they involve high current devices such as your welder, and could damage your Arduino.
The solution is to power the Arduino from a supply that is not connected to mains ground. Examples include a laptop computer running from battery power, a battery, or a 2-pin wall wart (in UK/Ireland, that would be a wall wart with a plastic earth pin).