Hi Charis and Mandy,
If I can do it you can too!
I spent 7 months backpacking solo in the following countries:
Peru, Bolivia, Chile (inc. Easter Island), Argentina and Brazil.
My advice would be to go with the idea that things can happen but may not necessarily. If you prepare yourself for something like being pickpocketed, then you wont fall apart when it does happen.
I have a few minor mishaps, but they've only all become great travel stories at the end of the day
I could tell you about them, but then they might only serve to freak you out, and really there's just no need to. If you are a smart traveller and you dont put yourself in harm's way intentionally or stupidly, then odds are you will be absolutely fine!
There are a few simple things that I stuck to when I was travelling:
- Dont walk alone at night. (and to be honest, this rule I try to apply to even my home country!)
- Dont miss out on really experiencing the culture just because youre travelling solo - so team up with other people from your hostel when you want to go out at night
- Dont flash expensive electronic equipment. That will definitely make you a target
- When you are on the move, split your credit cards into two seperate places, have a reserve stash of money somewhere safe, make a photocopy of your passport to keep in your luggage and keep your actual passport on your person. That way, IF the worst happens and you have to hand over your wallet, you have a back-up system.
- Dont get nervous if the South American men pay you a lot of attention. It can be intimidating at first if you come from a country where the men generally ignore you, like Ireland
. Actually, the vast majority of the South American guys are just being complimentary. They dont really mean to make you uncomfortable or offend you. I think this was something I learned as time went by.
- Use locks on your luggage.
- If you are travelling overnight by bus, dont leave your backpack on the seat beside you. Keep it firmly between your legs, with one strap anchored to yourself or something else, so you can feel if it's moved at all.
- Get out your Lonely Planet before you arrive at a bus/train station and try to orient yourself before you get there. There's nothing that screams "tourist" like standing bewildered at the door of the station with your map out trying to figure where to go next.
A few backpackers I met had some story about being pickpocketed or having a bag stolen, but the vast majority had nothing like this happen at all! 
There are plenty of solo backpackers down there and you will find that you are hardly ever alone unless you want to be.
There's no need to book hostels every time way in advance. The only time I reserved a hostel was if I knew I was getting off a bus somewhere in the middle of the night. Other than that, just go with the flow and see where your travels take you. 
Hope that helps. Travelling South America is an amazing experience, challenging on occasions, but totally worth doing solo!