A bad battery could cause alternator failure if the battery is at low voltage permanently and won't take a charge.
But usually a battery just being bad shouldn't cause alternator failure but in some cases it could and if your battery is bad you should get it replaced before you're left stranded somewhere.
If possible it's a good idea to replace car batteries every 3 to 5 years before it actually fails and leaves you stranded somewhere.
Alternator damage typically happens due to bad diodes or the alternator overheating.