I went by myself for a weekend to Rio and as a mid-30's female still did not have any issues. You just have to use all your common sense as you would at any tourist destination, or in fact in any big city. Especially at night.

I had a guide for a full day that took me to all the major sites, and we made many, many stops - it was awesome. This was for ease of touring instead of doing it all on my own primarily, safety, and it was not too much more expensive than a giagantic bus tour and I got to do a lot more activities too. The next day I walked miles from one end of Copacabana beach all the way through Ipanema beach and back, relaxed on the beach some, ventured off a couple of side streets to look into some shops and had no issues and did this by myself. I think one person approached me asking for $$, I just said no and continued walking, and that was it. I only had id, some $$, and a small camera that slipped into my pocket.

Ask at your hotel about any areas at night that you should not venture out into, or better yet, take a taxi. Likewise, you should be 100% fine if you do not shout in your appearance, 'I have money, come and rob me!'