South America is a vast continent and it would be easier to enumerate the places that are dangerous than those that are safe. It is true that violent crimes are more common than elsewhere but the majority of the continent is safe.
Crime rarely occurs in rural areas and so, even in a country like Ecuador where violence is rampant in many areas of Quito and Guayaquil, you can be completely comfortable elsewhere. A great example is the Galapagos islands which live off tourism but also trails in the Andes and coastal towns. Anything at the border of Colombia should be avoided though because there are armed militants and people do get caught in the cross-fire.
Peru is extremely touristy and, while there are some parts of Lima to avoid, nearly every place of interest is safe and patrolled by a dedicated and friendly tourist police, at least from morning to sundown. You do have to keep your wits about because crime does occur and pick-pockets are common in tourist areas.
Bolivia in general is also very safe for tourists. People are friendly and even La Paz is safe at night and the rest of the country even more so. As always, it is best to avoid dark and desolate places and if you keep in major roads, transportation and near places of interests, your chances for encountering crime are low.
Really, if you intend to visit historic centers and do outdoors type activities such as hiking, rafting, etc, I would expect you to need no more precautions than anywhere else. Argentina and Chile are even safer.
While I have not been to the rest of South America, I would still avoid Columbia for now, possibly with the exception of Cartegena in the north. Kidnappings, of foreigners in particular, is much higher there than elsewhere.
Added Brazil last year (2017). It is a huge country so I only covered the central part, just south of Rio de Janeiro at Ilha Grande to Recife in the north, plus Fernando de Noronha. The crime rate in Brazil is quite high but - as with other South American countries - is very unevenly distributed.
Rio is a big city with a lot of tourism. The areas near the main beaches, Copacabana, Ipanema, Leme is packed with people all day and easily until 11 PM. There is no problem walking there with camera in hand. It is a famed area for pickpockets though, so do keep an eye. Central Rio feels more dicey and gets barren at night, so it does not feel safe. Avoid the favelas which are known to be very dangerous.
Ilha Grande is relatively small island and pretty much only tourists go there. It felt safe day and night. A good number of beach towns line the coastal road from Conceicao de Abraao to Salvador de Bahia. Most of these are busy during the day and so feel safe. There is a decent presence of police, so just stay where people are to be safe.
Salvador is ranked at the 14th most violent city in the world! It really has a high rate of violent crime. However, it is possible to still see the touristy area. The historic Pelourinho is walkable until past 10 PM and there is a strong police presence inside but also at every read entrance to the zone. Do not walk past 11 PM I was advised. During the day, I covered the area several times. The other area of interest is the lighthouse and you basically take a bus to a zone of a dozen blocks surrounded by armed police. Once in, it feels safe with thousands of people bathing at the beach and staying until sunset.
Beach towns from Salvador to Recife are similar to those in the south. The small towns feel safe during the day and crime does not appear to be a major problem but the trick again is to remain in places where there are other people. Brazi is very hot, so many people go out late and stay late, so you won't feel forced to turn in too early.
Recife does not feel so safe but points of interest are in busy areas which provide relative safety. Olinda to the north has a restricted area around its world-heritage center and I was told you are more safe there than in a bank! Seemed like it, but it is not safe even just outside the area.
In contrast to all this, Fernando de Noronha could easily be one of the safest places on Earth. The entire archipelago is small and most of it is located within a marine reserve, so it costs to enter and stay there which gives the entire area a small-town feel. Everyone talk to everyone and I was given free rides by people who just want to help and chat with visitors.