There are a lot of nations in South America, each with a complex history, unique natural resources, terrains, etc. That makes it difficult to explain them all as “here’s why these countries all struggle.” However, if you’re trying to explain South America in a single narrative of why struggles with crime and economy are so common there are some factors you can point to. The most important uniting factor would be a lack of government stability. Controlling crime and creating a stable set of laws is paramount to continued economic growth, especially in an increasingly globalized economy that relies on steady trade and in many cases foreign investment. The difference in government stability can be connected to the legacy left by the parent countries that colonized the Western Hemisphere. That legacy can be characterized both by the function of those parent county’s governments, and also their organization and ultimate goal of their foreign colonies.
North America was colonized most successfully by the British. The English have a long history of rule of law (that is to say law is regarded above a single man and even the monarch must obey certain laws). Looking at events like the English Civil War or any of the myriad of important English legal documents (Magna Carta, etc.) you can see that their has been a long legacy instilled in English peoples regarding the importance of rules and laws. Furthermore, the goal of colonizing North America was based much more on setting up a permanent agriculture base and trading routes. This created a legacy in North America of a society that was here to set up roots, and had generation after generation of values based around representation of government and some semblance of equality (albeit a much smaller franchise of equals than we have today).
South American countries were mainly colonized by Spain. Although at the beginning of colonization, Spain was not so much one nation, but rather a conglomerate of nations with a single (sometimes 2) monarch. Furthermore Spain was defined by rule by law, meaning there are laws by which the people are governed, but those laws don’t supersede the monarchy. Essentially it was much closer to absolute despotism. There were also major differences in Spain’s strategy with their western holdings. Spain was interested in mining silver and trading in the East, but were less interested in (and even intimidated by) creating a self sufficient stable foreign economy. Their style of governance in the new world was to create small states with absolute rulers and no representation of the people whether they be descended from the Iberian Peninsula or native. These factors led to a population that was less accustomed to stable representative government and much more comfortable with the idea of a single dictator. It also led to a system of corruption and payoffs becoming normalized in society. Being that representative government’s stability is largely determined by the faith those being represented have in their government, it has led to natural instability. This is similar to the struggles we’ve seen in the 21st century trying to sell democratic republicanism to cultures that have no history of it. Simon Bolivar worked hard to create a centralized federalist system, but even he fell into the trap of despotism several times over his life while trying to keep his unified Gran Columbia stable. After Simon Bolivar died, Gran Columbia and the rest of South America began splitting into more natural boundaries and separate nations, many of them adopted constitutions based on the US model. Unfortunately many of those nations were defined by coups, corruption, revolutions, civil wars, etc. This behavior largely has not changed and can still be seen today. Whenever an elected government becomes corrupt or struggles to govern, people are all too willing to support these coups and dictators in the hopes that it fixes their broken system and brings some stability.
TLDR; stable government matters, a legacy of stable government creates legitimacy in government, which creates a more stable government and less crime.
Edit: formatting
And building on what bourbonsoakedscholar mentioned regarding there being a large number of countries in South America with different histories and outcomes, we can find countries in the region that have done quite well for themselves, albeit perhaps not reaching the level of economic development seen in the United States and Europe.
Costa Rica comes to mind as a great example of avoiding militarism and dictatorships, establishing a stable democracy, high levels of education and literacy, good environmental stewardship, and have placing 63 on the United Nation's human development index.