Is it okay to hear the fuel pump prime at ignition but does not cut off?

You're describing normal operation. The automatic shutdown, (ASD) relay turns on for one second when the ignition switch is turned on, then it turns off. It gets turned on again during engine rotation, (cranking or running). The Engine Computer knows that by the signal pulses it receives from the crankshaft position sensor and the camshaft position sensor. On an '88 model, those signal pulses will come from the pick-up assembly in the distributor.

The fuel pump gets turned on in one of three ways, depending on year and model. It can get its 12 volts from the ASD relay, a separate fuel pump relay can get turned on by the computer at the same time that it turns on the ASD relay, or the fuel pump relay can be turned on by the ASD relay. Regardless how it's done on a certain model, the fuel pump always runs at the same time the ASD relay is turned on. That relay sends 12 volts to the injectors, ignition coil(s), oxygen sensor heaters, and alternator field.

The purpose of this system is safety. All manufacturers use a version of this design. If a fuel line is ruptured in a crash, and the fuel pump keeps running, it would dump raw gas on the ground, creating a huge fire hazard. Instead, with a broken fuel line, there can't be any fuel pressure. With no fuel pressure, no fuel can spray from the injectors. The engine will stall. When the engine stops rotating, there will be no signal pulses from the cam and crank sensors, or the distributor pick-up assembly. The loss of those signal pulses tells the Engine Computer to turn off the ASD relay and fuel pump or separate fuel pump relay. That eliminates the fire hazard.

The only thing you weren't clear on is when you said the fuel pump doesn't cut off. Is that with the engine running or with the ignition switch in "run", but the engine is not running? If the pump runs anytime the ignition switch is in "run", including when the engine is not running, someone bypassed that safety system. That won't damage the pump, but it will keep it running if a fuel line gets broken. The fuel pump must run non-stop when the engine is running, but it must stop when the engine stops.