Volkswagen Repair: VW Jetta MIL, malfunction indicator light, vw jetta

Question
I bought my daughter a 2001 Jetta 24 months ago.  (4-Cylinder, auto trans, non-turbo with 102,000 miles now.) The battery went flat 9 months ago. Replaced it and then the new one went flat at the airport after sitting for two weeks. Blamed it on anti-theft system and radar detector left on. Jumped it and drove home.  

Battery started being drained after one week and then sooner so I replaced the battery under warranty.  Now the new one (3rd battery in 9 months) is acting up. Degenerated to the point where it won't hold charge overnight. Alternator still puts out 13.4-13.7 volts when running. Mystery current drain?

Today the Malfunction Indicator Light (MIL) stayed on after the engine started.  Won't go out.  First real clue!  Seems like something electrical is wearing/worn out.  Haven't checked error codes yet.  Need reasonable advice so I'm not "sold" a new auto-feedback-flyback widget that doesn't fix real problem.

Heard rumor that alternator can still put out voltage while causing mystery current drain when not running. True/Not True?  What else can I suggest to my mechanic before I start buying new parts for R&R process?


Answer
Paul,
  This is not my strong suit.  You might begin by observing the fluid levels in the battery (high amp charging will boil the fluid away in a short time, causing the battery to fail).  You will want to determine the amperage that the alternator is producing, not the voltage (which I believe, is more an indication of battery condition).  You will then want to put a portable ammeter (regular or induction model) on the positive side battery cable and read the draw down when the engine is both on and off.  If you find a large current draw when the engine is off, start removing suspect items from the circuit by unplugging fuses and then checking the draw.  You can determine if it is the alternator by unplugging the alternator. I have to mention that I don't know how this will affect a computer controlled circuit, so DO NOT make the intrusive checks without consulting someone who understands the ramifications of deleting systems from the circuit.  Hope this helps.  Jerry