If your girlfriend gets into a wreck while driving your secondary car are you responsible if you do not have insurance on that car?

Yep. Both of you are, in fact: her as the driver, and you as the owner.

And if your state requires you carry liability insurance on vehicles you own, you are particularly susceptible to an injured party coming after you.

To make a long story short, you allow your girlfriend to drive a vehicle which you know has no insurance. By doing so, you've accepted a pretty big risk. By accepting that risk, you put yourself in a dangerous situation if an accident occurs. Definitely something to think about...