Not necessarily true - if the value gets low enough it'll get the boot from the NYSE.
I bring this up because of the stock market crash having such ridiculous effects from the great depression era. Though I may just be uneducated on what really happened.
However, the owners of the company might get together and force the board of directors to do something, like fire the CEO. The CEO, who probably has a lot of compensation as stock or options, would like price to be higher, so he is likely to take actions to get the stock price higher. Such actions may or may not be wise, and may or may not be helpful for the company. Frequently they aren't...
Tell that to the employees holding options. Those options are likely worthless if the stock price goes down.