How do you know if your BMW E38 power steering pump is broken?

It is quite possible that the pump is OK and more than likely it is a suspension problem. Do you have an IL model? They have a self leveling suspension and the Power Steering pump also powers the suspension. It is a tandem pump using one oil reservoir. if the suspension is leaking then is would be hard to steer the car from fluid loss. Use only Pentosin fluid "expensive". I had a 735il and a 740il both develop steering problems. The 735il had the lines from the front to the back corrode at the body mounting points causing a massive fluid leak and the 740il had a flex hose in the rear leak. The steering is so hard you can hardly turn the wheel to park the car, it is a little easier driving along when the car is moving. You can drive the car to a service center but keep the trips to a minimum. It is not good to run the car with no oil as eventually the hydraulic power-steering pump will fail from being dry and when you fix the suspension problem you have created another and will have to replace the pump. Other causes might be the steering rack is leaking, or lines to it. Low fluid in the reservoir, broken power-steering belt, the pump really is bad (usually they make a noise) Fill up the reservoir and look for spots on the ground from oil leaking. If all the power steering parts are tight then it could be a bad steering box or front end parts like ball joints etc. I don't know how hard your car is to steer. With the 740il above it took two hands to steer the car jerking the steering wheel to get some leverage to get the car to turn. When everything is working you can steer with one finger! Hope this helps.

Don B