How does my breadboard circuit diagram look? Is my circuit going to explode?
You're not going to explode yourself, but your circuit isn't going to work.
According to your wiring diagram, your transistor is connected wrong.
Your control signal from the Arduino (D4) is connected to the emitter of the TIP120, while the base is connected to ground and the collector is connected to the motor.  It won't work that way.  Luckiliy, you have a 1k resistor in series between D4 and the emitter so it won't burn out your Arduino.
A proper low side switch looks like this:

Note where the base, emitter, and collector are connected.
Besides the transistor, you'll want to consider the current it takes to operate the valve.
According to the info page for the valve you linked to, the valve uses 15 watts at 12 volts when activated.  That's over 1 ampere of current.  Depending on the quality of your breadboard, that may be too much.  The contacts might get hot and no longer properly hold the wires.  Worst case you melt the plastic on your breadboard and ruin a bunch of contacts on your breadboard.

It would have been nice if you had posted a schematic diagram as well as the wiring diagram.
In this case, the wiring diagram was OK because you are specifically asking about the breadboard.
It's easier to discuss the circuit itself if you have a schematic diagram with part designators.  It is much easier to say &quot;R1&quot; rather than &quot;that 1k resistor connected to the base of the TIP120.&quot;