If the adult drops everything and runs screaming in panic for every small cut, then so will the child. That is no way to keep a child from screaming about cuts, scrapes, and other injuries.
As the adult, it is your responsibility to calm and soothe a child in a stressful or painful situation, not to make it ten times worse. Even if you are scared, worried, or nervous, the best thing you can do for the child is to never allow the child to see how you feel. A better thing to do is to aid the child with whatever first aid is needed, all the while acting like and verbalizing that everything is okay. Let the child know that he or she will be just fine, even in an emergency situation.
Take all the precautionary steps necessary, but without upsetting the child about cuts, scrapes, and other injuries. Do not let on that something might be wrong. You can call 911, if needed, obviously, but still reassure the child. Do not ever let the child see you panic. If it is an emergency situation, sometimes stress can worsen certain conditions. It is always best to keep the child calm.
If you cannot keep calm during the minor situations, such as cuts and scrapes, neither will the child. Children learn from what they observe. If a child gets worked up over a minor situation, imagine how panicked the same child could be over a big incident.
When I was maybe around 18 years old, I knew a girl that always panicked for minor cuts. She would act as if she were going to die. No matter how many times someone would attempt to soothe her, it never worked. It would take her over an hour to calm down. I always worried that if she ever had anything serious happen to her, she would put herself into shock. I'm not sure if that's possible, but I know she would likely get a nervous breakdown. I'm not sure if anything bad happened to her, but I sure hope it didn't.
The reason for bringing up that girl is that I remember her mom doing the same. If she tripped lightly, her mom would rush to her and say, "Oh, honey, are you okay?" Even if she said she was, her mom would still continue asking again and again and offering her band-aids and ice for something that wasn't even there. She would scream and panic. This mother's behavior likely led to the child's behavior. As I said before, children learn from observing those around them, especially their parents.
This is how I suggest handling minor situations. If the child just trips, a quick "Uh-Oh" and a laugh is really all that is necessary, especially if the child isn't hurt. If the child is hurt, still do the same and examine the child in a fun way, like mentioning that you want to search for Elmo or some other funny thing. Look at the child's injury and determine what needs to be done. Then, do it, but make it fun and say things like "Oh, that doesn't look bad at all. Let's just put a band-aid (or whatever other treatment is necessary) on here in case. "
Doing it in a fun way not only helps the child forget what is going on, but it also reassures the child everything is okay. After all, it must be okay if the adult is laughing, smiling, and playing.