G.R.
I was a cautious child--the other children were always too rough and too loud. I could always see consequences a few steps removed. My parents tried a number of things, the best of which was just letting me be myself. She will probably grow out of a lot of it, but you can't change her personality.
I suggest talking it over with her. Ask why she doesn't want to do stuff and really listen to her answers. Try reflecting her answers back to her, but phrasing them as positives not negatives. It will help her accept her feelings and move on, maybe she will even accept her carefulness and become a little braver.
For example: Mommy, "Why don't you want to go down the slide?" Girl, "I don't like it." M, "What about it don't you like?" G, "It's too high." M, "When you're up high, you worry about falling?" G, "Yes." M, "It's okay to worry. You are someone who thinks about things. Smart people can see a lot of possibilities. But if you want to and are careful, I'm sure you can do it. Being careful doesn't have to stop you from having fun."
But if she doesn't want to do it immediately after this conversation, don't keep trying to persuade her to do it. Sometimes it takes a little while for ideas to seep in. She might feel self conscious right away, but the next time or the time after something is scary, she might give it a try.