How can I stop supplying power to a circuit when power drops below a certain threshold?
If your LDO IC has a shutdown / undervoltage input, you can use that with a resistive divider to enable your IC.  What part number are you using?

If you don't have an input like that on your LDO, you could look into using a comparator circuit that looks at the input voltage and enables / disables your part through a logic enable line.  Hopefully your part has one of those?

This part might be overkill, but it has what I'm talking about:
http://cds.linear.com/docs/en/datasheet/3086fb.pdf

Also, if you're looking to power electronics from a vehicle, make sure you put some TVS diodes on the input - probably and SMA or SMC style package.  This will protect you from transients on the power supply.