Don't look and prepare mentally that you're not going to die there. I know it sounds absurd but the "fight or flight" situation kicks in and, like with panic attacks, the fear of actually dying probably leads you to pass out probably.
I have a friend who's girlfriend was driving one day and the car slipped off the road going towards an electrical pillar (it didn't hit it though because my friend managed to grab the wheel). Near the collision point with that pillar his girlfriend passed out, in an instant. We laughed at that situation for years. They ended up in someone's yard, killing a poor dog in his little house.
http://www.livestrong.com/article/166968-why-do-people-pass-...
However, that doesn't answer cuts. Does it happen on any sight of blood or only your blood?
I know a bit about this due to having researched after I gave blood the first time. Ends up the person was not very experienced and ended up drawing from muscle instead of a blood vessel. I was in for probably about 20 minutes with only half a bag before they gave up and sent me on my way. Walking from the parking lot to the office, I got lightheaded, nauseated, cold sweats, some gastrointestinal effects... But did not pass out. In my case, I think it's blood sugar related, as I had a similar reaction once while fasting.
Instead, start tensing muscles in your body - legs, hands, wherever. This keeps the blood pumping to your brain so you don't faint.
It absolutely works. I can even watch them draw the blood now and feel fine.