There are a couple of ways drivers go bad.
One is DC offset and/or heating. The signal going to a driver should be AC. However, some malfunctioning amps also send DC to the driver. DC means that the voicecoil will be constantly receiving power while connected. As you know, running electricity through a coil generates heat. Same reason transformers get warm, lightbulbs generate heat and why electric heaters work. The problem with voicecoils is that they're wound with very fine wire with a thin coat of enamel as an insulator. Heat it up too much and that enamel melts. When that happens, the coil touches itself all over and shorts out. The result is a dead driver. It won't make any noise at all. This is why it's critical to test amps for DC offset; they'll kill headphones.
The other major driver failure is when the cone itself gets damaged. This is pretty obvious, like when it gets torn. The surround can also get damaged and the cone can also separate from the voicecoil. Depending on the damage, it might still make sound. You can check for this by visually inspecting the driver. Most of the time the damage is obvious.
Though not related to the drivers, it's also important to check for damage to cables and the joints where cables are connected to drivers. These can flex a lot and go bad. A lot of times, it's just the cable that needs to be replaced or a solder joint touched up.
Which reminds me, it is also possible to kill a driver when soldering. DC offset destroys voicecoils because it generates heat. But so do soldering irons. If you're resoldering the joint where cables meet the drivers, too much heat can travel up the wire to the voicecoil and melt the insulating enamel. To guard against this, the joints should be soldered very quickly and the wire going to the voicecoil should have a heatsink on it. Even something like a paperclip can be used as a heatsink.