At its closest point the moon is 220 thousand miles away. At its furthest it is 250 thousand miles away. That is a difference of 30 thousand miles!
Clearly your airship is going to plow into the ground or fly off into space if you are not careful, there is a way around this though. At around the L0 point (so the weight of the cable is not relevant) you have a winching station. That winching station would spool in and out the cable as needed to keep it at the correct length and the airship at the right height above the surface of the earth.
However there is a much bigger problem:
The moon orbits the earth in 27 days (approximately). The circumference of the earth is 25 thousand miles, 27 days each of 24 hours. This would pull your airship through the air at 39mph.
However, the earth is spinning once every day. This pulls the airship in the other direction at 1041mph. In other words your airship is being pulled through the air at a combined speed of 1003mph, or considerably faster than the speed of sound at 761mph. Not only is this creating a sonic boom for anyone underneath but the drag and forces exerted on both airship and cable will be incredible. The cable will also be pulled a long way from vertical and have to be much longer as a result.