Base Jumping and sky diving are not dangerous sports relative to many other daily activities and other popular sports.
Many actives have some inherent risk, and if that risk wasn't there then we wouldn't do it.
For me, my activity is big-mountain skiing and ski mountaineering. I have lost friends, and frequently read about someone who dies, Certain situations have high risk but a high reward. Over 100 people died this winter in avalanches in the N. Hemisphere. Double that for ski related accidents.
We each have to choose the activity and conditions that have the right risk-payoff function. As Jean Boivin said, "You can accept to die doing this, you don't want to, but by being that close to the edge you understand what life really is, and it makes you a better person". I would rather die doing something I love than die in a car crash or die an 80 year old vegetable.
Your girlfriend must understand your desires and accommodate them. You must understand the risk and payoff and fit that in with the beliefs of your GF. I told my current girlfriend this, she does not like me doing dangerous sports but she knows she can't stop me doing something I love just like she knows she can't control me and stop me going tot he mountains.