How dangerous is passing out when moving from reclined seated to walking fairly quickly?

Injury. Passing out itself is not the dangerous part, you will regain conscious once blood returns to your head. The falling part that is dangerous. You can hit your head & cause serious injury. In this case, passing out occurs when you go from a seated to standing position too quickly that your body is not able to pump enough blood to your head (fighting more gravity when you stand up). Get up slowly.
Find out the cause. Depending on the cause, this can be fairly benign or very dangerous. It can be caused by a host of things from low blood pressure to anemia to a seizure disorder. You need to see your primary care doc right away for a work up. They may recommend that you see a cardiologist and/or neurologist before you're done.
Just be careful... Not too dangerous if you take it slow. If this is a new symptom, however, i would see your doctor without delay to get it evaluated.
Dangerous enough. Dangerous enough that you need an urgent evaluation. It could be from medications, heart rhythm problems or low BP from multiple other causes. In the meanwhile please be careful do that you do not fall and hurt yourself.