Control LCD with TIP121 on Arduino not displaying anything

With your original idea the LCD circuit is most likely finding a ground path along the other Arduino pins that are connected, and strange things are happening. Trying to enable a circuit by pulling down the ground point is usually not recommended except for simple circuits with components like relays, bulbs, or LEDs.

It is possible that the LCD system you have contains an enable pin that shuts it down. If available this could be controlled by an Arduino I/O port.

If the only alternative is to switch On/Off the power going to the LCD circuit you would have better luck switching just the "+" side of the power supply. Use a PNP transistor or a P-channel Mosfet (in which case switching is controlled by a low input signal). Before switching off the LCD it would be best to insure that all other data signals coming from the Arduino are set low. It would also be a good idea to check the data sheet of the LCD circuit or driver to see if dropping the "+" supply line is a potential problem. If the power supply to the LCD circuit is not compatible with the Arduino voltage there are choices of using an opto-coupled device (PNP or Mosfet type opto-coupler), or even a small low current relay.