How do I test fuel pump on 1993 bmw 325i. Car shut down while driving, will crank but not start.

post a problem first, not a hunt for things not there.
let me guess engine cranks but does not start and run and stay running , right <must we beg?
is fuel pressure zero 0- or too low , not even measured/?
no starts short list of huge.
1: fuses blown. we test those first.
2; bad fuel. (old or diesel over gasoline put in).
3: pump is dead or FPR bad and fuel pressure crazy too low.
4: no spark the PCM cuts fuel if SPARK fails by law, so does. if did.
5: scan tool shows DTC errors you ignore, even cranked?
CRASH CONTROL MODULE , DETECTED CUTS FUEL INERTIA SWITCH.? what the logic does here, I have no clue, ask a German?
6: ANTI-THEFT SYSTEM LAMP SHOWS noT HAPPY/?

Fuel Pump - can you here the pump (usually under the right hand back seat) running with ignition on?
Fuel Delivery (injectors) - squirt some fuel (just a cap full) into the manifold.
Spark - pull a lead off a spark plug and use a spare spark plug resting on the exhaust manifold.
Crank sensor - Injectors won't inject fuel if the ECU thinks the engine isn't turning. Crank sensor is at the front right lower side and as it gets older, the wire gets brittle and breaks most of the time where it meets the hard end. The connector is under the throttle body and should read 500-600 ohms.

How did you test the battery? The fact that you jumped it and drove home with no problem tells you the battery is the problem.

fuse? relay, check those in that order.

it definitely could be I would first recommend performing a fuel pressure test, also an old back yard test is to bang on the bottom of the fuel tank very hard rapidly while someone is cranking over the engine, alot of times this will make them work for a bit and start the vehicle

Get a DTC scanner and see what it says...in the meantime, check that fuel pump relay and fuel pump. The humming sound is probably the heater blower motor or part of the HVAC system winding down after disconnecting from power.

the problem is your sensor is bad . the sensor for for the fuel pump is not sending power.

check the fuel filter and if old replace it and if that does not work, change fuel pump and strainer.

I had this problem in my shop and its been a Crank sensor and on one occasion it was a fuel pump relay that was bad. Most like its a the cranks sensor that's bad NOT the fuel pump relay but check both. To check see what missing when vehicle doesn't start or shuts off . If there is NO gas its the relay. If there is NO spark its the cranks sensor. Good Luck and contact if you need more help. Thanks for using FIX YA

Ignition module is the likeliest place to look.