If a ship sinks, what does an airship do?

'Descend,' 'dive,' 'drop,' and 'fall' are all perfectly fine and, in the case of an uncontrolled descent, 'crash' does indeed tend to be the end result (though this refers specifically to what happens when it stops falling due to an undesirable form of contact with terrain.)

However, 'sink' is still perfectly valid to describe the actual falling. Indeed, pilots use the term 'sink' (usually when descending faster than normal) and use the term 'sink rate' to refer to the rate of descent of an aircraft.

From Wikipedia:

The rate of decrease in altitude is referred to as the rate of descent or sink rate.

Similarly, Ground Proximity Warning Systems on newer aircraft use the term 'sink' when they produce automated callouts. When an excessive rate of descent is detected by the radio altimeter, the following callout may be generated:

Sink Rate! Pull up!

When altitude is lost after takeoff or with a high power setting, the following callout may be produced:

Don't Sink!

Source: Wikipedia. Also, I'm a pilot (of small airplanes) and hear the terms frequently used by other pilots.

Very simple, it crashes.

This word has been used throughout the history of airships. Most famously the Hindenburg.

An airship is just another form of aircraft. It is not a ship of the seas. The vocabulary of flight is applied.

It plummets.

Plummets - A steep and rapid fall

It falls

It's not part of the plan for a ship to sink and it's often due to unfavorable circumstances. An aerial craft in free-fall is the closest equivalent I can think of. On an airship this would like be cause by a puncture or any other escape of the gaseous body.