If you witness a crime being committed, do not intervene unless you are really sure of what you are doing: many criminals, even pickpockets, carry guns, knifes, etc and may use them if feeling threatened.
In general, a tried and true technique for staying safe in Lima is to simply maintain a low profile. Leave the Rolex at home, don't wear the fine suit and don't carry a laptop when hailing taxis on the street, and keep a relaxed, friendly, smiling attitude. If you do need to go out dressed like a gringo, call a taxi rather than hire one in the moment - the few moments you wait and the few extra soles you pay will be worth it.
taxis
Another important point to be taken into consideration is that you should not pick up just any taxi, especially when you are leaving the airport. It is not strange to hear news that some taxi drivers cheated tourists for example, going from the northest point of the city to the southest part would take you at most 50 soles and that is the largest distance in Lima so do not pay more than that by charging them 100 or even 200 soles for normal rides even though Peruvian taxi drivers normally tend to increase their fares in front of gringos, it is not a massive difference. It is most advisable to use one of the official taxi companies inside the airport such as Green Taxi with set fares to ensure your safety.
Informal taxi drivers have also occasionally been known to participate in actual robberies and express kidnappings. While the overwhelming majority of Lima's taxistas are just poor working people trying to make a living, you should be alert if you are going to hail a taxi on the street, especially if you appear to be wealthy and/or a foreigner. Your safest bet is to have your hotel call a taxi for you or keep the numbers of official taxi companies "radio taxis", which are marked with registered numbers handy. Lima's tourist information centers will be willing to call one for you as well.
football violence
Avoid the surroundings of Soccer / Football stadiums before and after big matches, since "barras bravas" hooligans can be very violent. Ask for advice if you plan to go there or thereabouts. Very infrequently, but occasionally, even in nicer tourist areas, gangs of youths, sometimes from rival high schools, or supporting rival football clubs, or strikers involved in a labor dispute may brawl. If you find yourself caught in the middle of such a confrontation, just try to move out of the way, preferably behind a closed door - these youths generally do not carry lethal weapons, and the worst that is likely to happen is that someone will get hit with a rock before the police arrive to break it up.
sex
Staying safe for adults can also require an understanding of the sexual climate of Peru. In general Peru is a relatively conservative country in the sense of male and female roles, but at the same time Peruvians are extremely open to friendships with foreigners. Thus, some males can find themselves suddenly the object of flirtation by attractive young Peruvian women, but then be suddenly rejected for having violated some unwritten line of conduct in, say, discussion topics. Women can find themselves the object of unwanted looks and stares, but at the same time the risk of violence and rape is probably not as high as in many other countries.
A problem that can arise is the Peruvian concept of the pepera, found at certain night clubs or pubs. Peperas are usually attractive women aged 16-25 that deliberately entice foreign tourists and then spike their drinks with sleeping pills and rob them once they're unconscious. Usually peperas work in groups of two, although smaller and larger groups exist as well. Male "peperos" also spike the drinks of women but robbery is often accompanied by rape. Peperas in general are found in dense tourist areas, such as Park Kennedy in Miraflores as well as the Plaza de Armas in central Lima. One locale in particular that is notorious for dangerous peperas is the Tequila Rock discoteca in Miraflores and its sister in Pueblo Libre La Marina.
Another cultural concept worth learning is the "brichera" or "brichero". There are two types of bricheras: the first type are women that are genuinely looking to meet foreign men in the hopes of dating or marriage or even a quick fling. The second type are women that search for foreign men with the implicit purpose of exchanging sex for small gifts or money. This second type of brichera is risky, especially for foreigners lacking local sensibilities, since it involves prostitution. These bricheras do not use contraception reliably, and therefore pose a higher risk for transmitting STDs Sexual Transmited Diseases. If you decide to have a fling, make sure to use a condom.
districts of note
Some areas of Lima are safer than others: Miraflores and San Isidro have large populations of well-to-do and wealthy Peruvians, not to mention large tourist groups, so they have large police presence to protect the population. Other districts, such as La Victoria, are much more dangerous. Visitors would be well advised to stay out of these areas unless accompanied by an experienced native or visiting busy areas during daylight hours. Downtown Lima is normally well patrolled but be careful anyway. Callao the port, technically a different city is rather rough: ask for advice before going there if you plan to. The area around the airport is generally safe and well guarded but use common sense while lugging your luggage outside the airport.