Will This Solenoid Driver Circuit Draw Any Current When Control Input is Disconnected?
There will be some leakage through the TIP120.
There will be a small amount of leakage from collector to emitter, and some from collector to base (which will then go through the 100 kΩ resistor).
It may not be well specified.  A datasheet I am looking at for the TIP120/TIP122 shows about 0.5 mA from collector to emitter if Vcc is 30 V.  See page 2 HERE.
You can find something else to use with much lower leakage than a darlington switch.  Some opto-isolators, for example, could drop right in and be over an order of magnitude lower leakage.