I'm sure the radio draws from the battery even when the key is off. Otherwise, how does it keep your clock settings? But the power draw should be very small.
Here's what I do to check for a battery drain. DO THIS AT YOUR OWN RISK, IF YOU BLOW SOMETHING UP BY MISTAKE (ECM, MULTIMETER), DON'T BLAME ME:
1. Get a multimeter that measures current.
2. Set the meter to do so. It usually involves switching the probes.
3. Set the meter to measure 10 Amps or less.
4. Find somebody that has a similar mustang that does not have any electrical problems.
5. Buy them a coke/beer and tell them you'd like to take their positive battery terminal loose.
6. While the car is off, take the positive terminal loose and remove it from the battery. Be careful not to lay it on anything metal!
7. Place the meter in series with the battery by putting the red meter probe on the battery positive post and the black meter probe on the battery cable.
8. Make note of the current draw .
9. Repeat this on your car.
10. If the difference in your car and your friends is significant, like around 0.5 to 1.0 AMPS, you probably have something shorted out that is draining your battery.
If you suspect it's the headunit, pull it out and see what difference it make in current draw.
Black '99 GT , 3.73 gear, SpeedCal, K&N filter, Bullit wheels on Goodyear Eagle ZR 245/45, QA1 k-member, Granatelli front street coil overs, Steeda 555-8200R rear springs, Tokico struts, QA1 c&c plates, D&D Motorosports subframe connectors
'70 Fastback, under restoration