Electronic – LED current while off

First you need to detect the blinking/not blinking.

An RC lowpass filter followed by a comparator would do this.

Then you need to use the comparator output to switch between the LEDs.

I have thrown together a quick circuit that should work (sorry for the mess but I'm rushing at the moment):

And the simulation:

The pink line is the PC power indicator in, you can see it starts to toggle after around 2ms (I forgot to expand the time realistically, sorry - depending on the frequency of the flashing you will need to adjust R1 and C1 - probably 10k and 100uF are better values)

When it starts to toggle the voltage after the RC filter (RC_FILT) drops below V_REF and the comparator output switches (not shown)
Depending on the state of the output (5V or 0V) either the NPN or PNP transistor is on, and the LED in series with it is lit.
The botton graph is of the current through each LED - you can see one drops to 0 and the other turns on when the toggle starts/stops.

Hope this helps - ask if you don't understand anything and I will try to add some more later if needed.

EDIT - here is another version that does not use a separate power supply. It's a quick hack so I make no guarantees - the components shown are guidelines, you can use any small signal schottky and pretty much any small N-channel MOSFETs. This is about as simple as I think you can make something to do what you want:

Here's the sim:

You need the LED to be lit when the photoresistor is high resistance. So replace the photoresistor with a fixed resistor R3, to supply the base current to turn the transistor on.

Then you need the LED to turn off when the light shines, and the photoresistor is low resistance. So connect the photoresistor from base to ground.

Now, when its resistance is low enough, it will drain the current from R2 to ground, and hold the base voltage below 0.6V turning the transistor off.

Say, at 3kilohms we aim to get the base voltage down to 0.3V. Then 0.3V/3k = I = 0.1ma. Then R3 must drop the remaining voltage 4.7V at 0.1ma, so R3 should be 47k.

Now the transistor will start to turn on when the photocell resistance exceeds 6 kilohms. If that's still too bright, increase R2.

Answer in a schematic