On 280ZXs, the fuel pump comes on when the ignition goes on, so you will always hear the fuel pump before the car starts, even if it's a real quiet one. On 280Zs, the fuel pump only comes on when the engine is cranking, so you won't always hear it until the engine is running. In either model, fuel pumps are usually audible when the car is running. As fuel pumps age (actually, usually right before they give up the ghost) they do often get louder. This is usually accompanied by symptoms like hard starting or lack of power.
So just hearing the pump going when you turn the key on your '83 does not mean it's going bad. However, if the pump is much louder than it has been in the past, imminent fuel pump failure is a possibility.