Other than carefully using a good multi-meter to test output, and see if the power
supply is within the stated (or otherwise published?) parameters of the display,
you may not be able to tell directly. Sometimes, there could be another part inside
the display itself that is acting up; at least the power supply is external, accessible.
There may be a few youtube or other online videos, that show some aspects
otherwise more difficult to fathom; you may have to spring for a $100/$200
replacement supply by part number, however if the circuit board inside the 30-
inch display has a failed component, then you'd be looking at other parts, too.
For a look into a cross section of retail/used displays and a few components:
http://www.thefind.com/computers/browse-apple-cinema-display-power
Someone adapted a different power unit from something else, here:
http://pixelchimp.net/blog/pixel-chimp/replacement-power-supply-apple-cinema-dis play
There are motherboards and other parts inside, to include cold fluorescent backlights.
OR with later model, the LED. And all early models did not have MagSafe DC outputs.
And these can add to the troubleshooting. If the display worked with a different Mac
then didn't with yet another, but returned to functional status when swapped back to
the original or another; this may indicate a faulty wire near a plug on a cable. Or it
could indicate something that was running hot (in display, or in specific Mac) that
shut off due to built-in limits and temperature sensitive overload protection.
Not sure if the iFixit site shows any inside information on early or late Aluminum Cinema
displays; first three, as stated in an older Apple Support article may have different parts.
The later model included ThunderBolt2, while earlier build didn't.
If the power supply has the input/output specifications, then you may be able to check
the output against a good digital multimeter. Or carry the unit to a shop (call ahead) to
see if they can test it for you. Someone who has a known-good power unit certainly
would be a very helpful person to know. Some of these were backward compatible to
a smaller screen model; but not all. The small 20-inch used much less output, so the
limit on that may be to use a 23-inch adapter with the 20-inch; or 30-inch with 23, etc.
The newer models may not share the same power supplies. I have not researched this.
Older ones do not have the power for MagSafe to use with MacBook product. When
replacing the power supply be sure there is adequate compatibility; a slightly higher out-
put supply should be OK since it won't under-power the display, other things being equal.
(Or, if not, that may be the rub.)
So, if you have the meter to test power output, that may be a start.
However with the circuitry inside, other troubleshooting is likely.
With the DC output from the display adapter, a good meter should
be simple enough to test. It would show a negative value if you do
not have the leads in the correct power source for + pos - neg.
Good luck in this matter...!