I am rewiring my old finished basement and laundry room, but some of the existing 50+ year old BX branches off to the upstairs. Also, some of the wiring is controlled by 20 amp breakers and some 15 amp breakers. Is there any problem with just using 12/2 wiring throughout the project or must I use 14/2 for any line that is controlled by a 15 amp breaker?

You can use 12/2 throughout.

12/2 15 or 20A breaker
14/2 15A breaker

KISS is correct, 14/2 is the minimum wire on a 15 amp circuit but you can always go with heavier wire.