It sounds like what you need are a couple of stepper drivers with an enable. Feed the same four lines to the inputs of the drivers, and connect one stepper to one driver, and the other stepper to the other driver. Then feed the enable pins to a couple more Arduino outputs -- then use those outputs to enable or disable the drivers. Probably better to use a driver IC, rather than trying to do this with transistors.
Google: "stepper motor IC" or "Stepper driver IC"
And, I'm thinking if you can drive those steppers with an Arduino output, then probably any driver will do the job. Just make sure it has an enable or chip-select or the like. you might be able to find a dual driver that will do the whole job. Also, look at "break out" boards and stepper driver modules.