How are fire trucks supposed to respond to incidents involving airplanes, as it appears this case involves, if the runway is off limits to them?

The way it's supposed to work, the ground controller first verifies that there are no traffic conflicts before clearing vehicles to cross an active runway.

Which is exactly what failed here, so saying "it shouldn't fail by not failing" doesn't help terribly much.

Having grade-separate crossings for vehicles might, but that introduces new issues (plane skidding off runway could hit the incline and break up).

O’Hare has those but it’s not helpful for emergencies that happen on the runway itself.

Well, sure, but in that case it's expected that the runway is closed.

The fire truck was responding to an emergency which is why it needed to cross an active runway.

That is exactly my point. What visual aids do the ATC controllers have at their disposal to decide if the runway is free for an emergency vehicle to pass?

If it's not safe to use, then they should wait or go around. Otherwise accidents like this one happen.