I have an effects unit that is switched on/off via a mono jack cable with a mechanical footswitch to complete the circuit. I don't think any voltage flows down the cable. I can use a relay, but wondered if there was a simpler solution? Is it possible to use an analog pin as an input then connect it to ground via coding? Perhaps a transistor switch would do the trick?
When you have an unknown circuit a relay is the best solution because a relay is an electrically-isolated, electrically-controlled, mechanical switch. Sometimes you can use a solid state relay, but you have to use the "right" solid state relay (for example AC & DC solid state relays are not interchangeable).
If you know a signal is pulled to ground or to 5V (etc.) by a switch, you may be able to use a transistor or MOSFET to replace a switch.
Is it possible to use an analog pin as an input then connect it to ground via coding?
You can connect an input to an existing circuit as long as the Arduino's voltage limits are not exceeded (nothing greater than +5V and no negative voltage), so for example you might be able to read the state of the existing switch. But, you cannot generally connect an output to an existing circuit (or you cannot connect two outputs together), so you wouldn't be change the state of an existing circuit under software control that way.
Put a meter on the two pins that are opened/closed by the footswitch. Possibly +9V (battery+) is being connected/disconnected, or the battery - is being connected/disconnected.