I do not see any indication that the Adafruit driver you linked to has any built-in current limiting features, which means your operating voltage should not be any higher than what you stepper motor is rated for. As stated on the overview page you linked to “You can use any DC or stepper motors that run from 4.5-13.5VDC and draw under 1.2A per coil.”
The #1204 stepper motor has a rated voltage of 3.9V. Since this is too low for the operating voltage of the driver, the #1207 stepper motor (with a rated voltage of 7.4V) is probably a better fit.
As far as selecting a battery, you should make sure it does not exceed the voltage rating of your motor (as mentioned above). With the #1207 stepper motor, a pack of 5xAA NiMH batteries might be an option. You might also consider using an adjustable voltage regulator like our S18V20ALV Adjustable 4-12V Step-Up/Step-Down Voltage Regulator to give you more power supply options.
Alternatively, you might consider using one of our stepper motor drivers. We carry a wide range of dedicated stepper motor drivers that offer a variety of interface options, all of which feature adjustable current limiting: