How to quantify the health of an electronic appliance according to its power consumption?
Well, if the device is supposed to draw only 500mA, but it keeps blowing a 16A fuse in your house, then the device is probably busted.
Outside of that, you'd have to know (or track) typical power consumption for the device.  If it draws significantly more (or less) current than usual then there may be something wrong with it.  If the current consumption slowly increases, then it may be going bad.
In truth, it will depend on the device and what you know about it.
You won't (generally) be able to put a device on a power meter and say "well, this thing's gonna go pop next week."  Any information you get will have to be seen in light of what the device is expected to do and how it has performed in the past (or how devices of the same model generally perform.)