Red wire on the shift solenoid

There are multiple red wires. The one feeding the solenoid pack gets twelve volts from the transmission control relay. That gets turned on at the same time as the automatic shutdown, (ASD) relay turns on.

What kind of problem are you trying to solve?

The Engine Computer turns on the automatic shutdown, (ASD) relay for one second when the ignition switch is turned on, then again during engine rotation, (cranking or running). That relay sends current to the ignition coil(s), injectors, alternator field, oxygen sensor heaters, and fuel pump or pump relay. The computer knows the engine is rotating by the signal pulses it receives from the crankshaft position sensor and the camshaft position sensor.

If a fuel line is ruptured in a crash, there will be no fuel pressure, and the engine will stall. Once stalled, there will be no signal pulses, so the computer will turn the ASD relay off. That stops the fuel pump from dumping raw fuel onto the ground where it would create a fire hazard.

If the engine runs, the ASD circuit is working, and there's nothing to check there.

You asked how to check the ASD relay. I told you it's working if the engine runs, and I added why it's used and how it works. It is turned on at the same time as the transmission control relay, and by the same circuit. The red wire you asked about gets 12 volts switched onto it by the transmission control relay.