Ha ! Ford introduced that system in many of their recent cars - 'Smart Charger' - they call it, only it would have to be the dumbest idea on Earth if you live in a place like Australia, where it's very easy and not uncommon to drive for 3 - 5 hours (and longer) without using the Brakes. A Mate found this out after buying a Ranger and then heading off to Lake Eyre and having a dead battery before the day was out and it only got worse from there,,,
Anyway, we aren't talking about those kinds of situations here.
The advantage of an electric pump is it can run in concert with water flow requirements rather than directly proportional to engine speed, which is not at all related to engine heat and coolant flow needs.
I think the 1 Amp figure Curely quoted might have been with the Pump unloaded? If you made a closed circuit, with a valve in it and run it again so you can adjust rates and pressures with a restriction in the system, I think you'll find more real life current draws. 1 amp at 12 volts is only 12 Watts. Ignoring losses here, that's only 0.018 HP. The stock pump with 3.75" Pulley fitted drags about 2 HP from the the Crank in a 'peak loading' situation, where the Pump is spinning at it's optimal speed which for these pumps (depending on which one you have) is usually between 2000 and 3000 Pump Shaft RPMs. Above and below those speeds the power needed is less and also, the power needed at any speed will only be (kinda) proportional to any restriction of the pump's (capable) flow rate. If it's open ended, then the power needed is going to be next to nothing and as they are a Centrifugal Pump, even when totally shut off is not when they are loaded the most, that occurs at is optimal operating RPM and peak flow rate (as will be set by restriction in the system). It's not at all straight forward.
Given the small numbers of power involved here, for a street car, I feel it's not worthwhile if only considering perceived power savings, which may or may not occur. As some guys touched on, while it won't be directly driven from the Crank, it actually still is, only via the Alternator. If the losses were nil, the power absorbed to run the Pump would be the same, but as the Alternator and the Electric Motor of the Pump will have losses, the power is in fact more.