My AC and Heat systems are separate, Can I wire one or both fan(only) controls to my thermostat?
You can certainly connect and use the fan-only wire for the AC.
Whether you can also drive the heating unit's fan-only input depends a great deal on your configuration and is probably not a good idea.
A breaker popping for one fan is not a concern for the thermostat system as the controls operate via 24V relays.  As long as the appropriate 24V transformer is still powered, the other fan/system will operate normally.  No additional harm will accrue to the dead fan other than what caused the breaker to trip in the first place.

First you must determine your exact control configuration, especially if you have one 24V transformer or two.

Provide a picture of your thermostat's base-plate (wiring terminals).
You should have 1 to three power terminals, typically labeled: R, Rc, and Rh.  Is there one wire connected to these or two wires?
Is there one Common (C) terminal or 2?  With 0, 1, or 2 wires?
Verify you only have one fan-relay terminal, typically labeled G.

The biggest problem with one thermostat for 2 separate residential systems is usually the power supplies. (2 different 24 VAC systems tied together often leads to one 0 VAC system and possibly smoke or a blown fuse.)
Your thermostat is designed to keep these 2 supplies separate; trying to drive both fans at once can tie them back together (and release the aforementioned smoke).

If you have 2 or more power wires and/or 2 or more common wires,
then you have 2 separate 24V transformers (this is the normal configuration) and driving both fan wires will do one of three things:

Not drive one fan at all (usually heating) due to the thermostat applying Rc power to an Rh circuit, both un-closed, for the heater's fan relay. This is the most likely scenario.
Immediately blow one or both transformers and possibly one or both control modules.
Appear to work (transformers roughly paralleled) but transformer life will be shortened due to current loops and slight phase differences.

In this scenario, you would need to bust out your Electronics-engineer skills and add a low voltage relay to safely drive the heater-fan relay (and not blow any transformers).  That's beyond the scope of this question and there might be some commercially available solution an HVAC pro could hook you up with, maybe.

If you have only one power wire, and zero or one common wire(s),
Then the last installer might have driven everything off one transformer . And it might be a third transformer installed just for that purpose.
In that case, it might be okay to connect both fan wires to the same thermostat G terminal, see below.
The next question becomes whether the thermostat and transformer can supply enough current for the fan relays on both systems.
The thermostat can probably handle it, but the transformer might not supply enough current.
It would be prudent to:

Locate the 24V transformer(s).  Each air handler should have one, and one or both should be disconnected (in this 2-wire scenario).
There may also be a third transformer installed to meet the demands of the combined system.  In this case, you're probably good to go.

If labeled/possible, note the current or wattage the transformer is rated for.

Test the furnace fan control.  Unplug the thermostat and jumper the power wire to the heater's fan(only) wire.
Does the heater fan come on right away and the air stay cool after 10 minutes?
~~~ Do not attempt to run both fans just yet. ~~~

If the fan failed to blow cool air in step 3, then your heater may not really have a fan-only mode.

If you can, measure the current across the jumper in step 3 for both the heater fan relay and the AC fan relay (separately).   The total of both currents should be less than 95% of the transformer's rated capacity.

If the conditions of step 3 and 5 are met, then you should be able to connect both fan wires to the appropriate (usually G) terminal of the thermostat.  There is a small chance that it will burn out the thermostat, but it probably won't.