It is normal for alternator output to fall off at idle. Alternator output is proportional to how fast it is spun. If there is a heavy load on the electrical system (e. g. headlights, grid heater, big stereo amp) you'll see that the "voltage drops way down" when the alternator turns slowly, like at idle. (On my 1995, the voltmeter reads about 11 volts with the grid heaters on and engine idling. Completely normal).
The PCM uses engine RPM to shut down the alternator field when the engine is not turning. This conserves the battery -- the alternator puts out nothing at zero RPM, and running field current through it is a waste of electrons. The PCM may also engage in fancier control strategies using engine speed/TPS/etc. You could imagine it squeaking out a few more HP at full throttle by shutting down the alternator, for example.
No CPS => no engine speed input to PCM, and PCM leaves alternator shut down. That's not the original poster's problem, as the alternator does charge at higher RPM.
As DieselDoghouse suggests, a bad alternator diode will amplify the impact of engine RPM on alternator output. You can see the missing pulses with an oscilloscope on the main lug of the alternator (and actually the whole electrical system, though it will be most pronounced right at the alternator). The "alternator field" terminal is what controls the rate of charge, and is an input, not an output. It is not a good place to check for bad diodes.