The simple answer to that question might be "possibly". As voltage goes down, the amperage on a device would go up. The fuses are rated in amps, so there is the possibility that a weak battery may cause a fuse to blow.
If your charging system is doing it's job, I would not expect a weak battery to be the problem if fuses blow with the engine running.
If your fan motors insulation in the windings was starting to break down or thore wise short out the armature, amp would start to go up. Also bearing(s) in a motor going out will also increase amp draw. If a motor is somehow restricted in coming up to full speed or other increases in loading on the motor, amps would also go up. Might also want to check for restrictions in the air flow. (Plugged inlet or outlets of the heating/ventilation system.
