In your diagram, you have notated the transistor emitters as "-24V." This is not correct; you should show this point as ground. This same ground point should be connected to your driving logic's ground.
You should have +24V connected to the motor common wire, as you show it.
The transistor base resistors (R1, 2, 3 and 4) should be selected according to the transistor hFE and the collector current drawn by each motor winding. The general equation is Rb = (Vlogic-Vbe)(hFE) / Ic, where Rb is the base resistor, Vlogic is the voltage provided for a logic "1" state by your driving circuit, Vbe is the transistor base-to-emitter "on" voltage, hFE is the transistor's gain, and Ic is the motor coil current. So, as an example, given a motor coil current of 1A, a Vlogic of +5V, and an hFE of 1000 for the TIP142 transistor, Rb = (5V-3V) (1000) / 1A = 2000 Ohms. In these assumptions, I am ignoring the TIP142's internal resistors, which would use about 370uA of the base current, and I am using the Vbe(on) value of 3V from the Fairchild TP142 data sheet.
You may want to add back-EMF protection diodes across each motor winding, Type 1N4004 will work for a small motor using 1A per coil. These diodes will protect the transistors from the voltage induced in the motor windings when they turn off, and also possibly prevent inadvertent upsets in your driving logic. However, doing so may also affect the torque characteristics of the motor by maintaining winding circulating current. Also, add a decoupling capacitor from your motor's common wire to ground, such as a low-impedance 100uF, 35-50V electrolytic type.
If the motor still doesn't run, examine the logic sequence driving the bases. A four-channel oscilloscope would be ideal for this purpose. I hope this helps.