I'm late to this post, but i think I understand your confusion. First of all, when applying voltage to the bass of a transistor, (presumably with the emitter grounded),you have to remember that the Base to Emitter junction will look like a diode. A diode will behave like an open circuit until it sees about 1/2 volt, at which point it will change only slightly. In fact, if you try to force it much beyond .6 volts, you will fry it. But the BE junction makes sense if you think of it as a pure current device (or current "sink"), and the transistor itself as a "current" amplifier. So assuming you have a 5V source available to drive the BE junction, what you have to consider is "hFE" parameter, which translates into the current gain. That current gain will change depending on the load (its not a straight line), but its a good starting point. So lets say you have 5V, and we know the BE junction will try to maintain about .6V. Now this is a darlington, so we'll double that and assume about 1.2V. So that means you have 4.5 Volts to work with. Now since we're dealing with a pure current "sink", we can simplify say that if the TP transistor has an hfE of 1000 at 3 amps (its high, of course, because it is a darlington transistor). So that means all you need to drive it is 3 mA ( 3/ 1000). So, plugging that into ohms law, you have (5 - 1.2) / .003, = 1266 ohms. So all you need to do is feed the available 5V through 1266 ohms to the base of the Emitter grounded transistor, and you're good to about 3 amps. Add a little extra current to ensure saturation, and the 1000 ohms offered earlier should be perfect. But to be sure, recalculate based on your actual load.
Be aware that the darlington transistor will "eat" up a little over a volt from your load. Just like the BE junction, the whole transistor will lose some voltage across it.
Now... one more tip.... since you have 12 volts to work with, and have already committed to adding an opto idolator, you might want to make the jump to re-thinking this to work with a MOSFET. Google some basic MOSFET circuits and you'll see they are a lot easier to work with than you may think. believe me... I understand about sticking with parts and components you already have some familiarity with, and I too tended to use transistors (and darlingtons) in many cases where a MOSFET would have been better, and after making the switch once I was hooked! A MOSFET, first of all, won't "eat" a volt or so from your load. Instead, what you lose will likly be in the millivolt range. Second, since there's less voltage "dropped" across a MOSFET, it will dissipate much less power, run cooler, and control a much bigger load for the same size part. Want more reason to switch? A MOSFET (unlike a BJT) is a pure VOLTAGE device. You won't have to worry about how much current to drive its "gate" with, because for all practical purposes a MOSFET draws no gate current at all. You'll just need to throw in a pull up resistor, and you pretty much can pick the value at random. Have I sold you yet? The oly down side is that because they are a voltage device, and the Arduinno (like many processor boards) only offers 3.3V, finding MOSFETs that can work without an interim voltage amplifier (or opto idolater) will take a little more careful parts selection. But opto isolation is a good choice and solves a lot of protection problems, so as long as you work with those, you probably can get ideal MOSFETs for your application dirt cheap.
