The issue could be several things:
a) bad alternator,
b) alternator is good but not charging battery
You can easily find this out using a multimeter (if you don't have one, spend some money for a decent one [like $50 or more], it's a good investment).
Step 1) With the car off, battery connected, measure the voltage across the battery terminals. Voltage should be around 12V (a tad below is fine). If the voltage is a lot below 12, like 11.4 or lower, then the battery needs to be charged.
Step 2) With the car running, battery connected, again measure the voltage across the battery terminals. The battery is now being charged by the alternator so the voltage measured should be around 14V. If it is too high (like 16V) the voltage regulator in the alt is shot, alt needs to be replaced. If the voltage is too low, it's because either 2.1) the alternator is bad or 2.2) the charging circuit is faulty, so the alternator is not being told to turn on.
The best way to tell if the issue is 2.1 or 2.2 is to take the alternator to a shop like Auto Zone or Advanced Auto and have them test it. If they test it and say it's good, then your issue is 2.2 - charging circuit. I am familiar with this circuit and will be happy to walk you through troubleshooting it. If they say the alternator is bad, then replace it. You may also need to put your battery on a charger - the alternator canNOT recover a battery that has been deeply discharged.
To answer the question on why disconnecting the battery from an EFI car *might* be bad: the battery in this EFI system is used as a sort of buffer - if there are any surges or spikes, the battery absorbs it rather than delicate electronics. The reality, though, is that the alternator (and it's voltage regulator) is what is really controlling the voltage, surges and spikes. So, the battery should be able to be disconnected from this system with no problem. On a car that is running normally and fine, then yes the battery can be disconnected. Shouldn't matter. However, if something goes wrong (for example a master cut-off switch is hit) and the voltage all of a sudden has no where to go (because there is no battery and no specific ground provided for this situation), then it will most likely ground out through delicate electronics, typically destroying them in the process. A carb car is not as susceptible as it doesn't have the delicate electronics. On an EFI car, with a properly designed charging system, removing the battery shouldn't cause an issue (though cars now may have sensors for this), though I wouldn't recommend doing this too often or for too long. Also, some cars actually measure the temperature of the battery to help determine certain things (like injector pulsewidths) but this is typically in startup conditions and not running conditions).
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