What were some of the factors that contributed to some South American governments being so sympathetic to Nazi war criminals that they harbored them?
In the late 19th and early 20th centuries there was a massive wave of German emigration to the Americas, the numbers are a bit fuzzy, but there's little doubt that at the conclusion of WW2 there were strong German speaking communities all around South America, mainly in Argentina, Brazil, Uruguay and Chile. Furthermore, the politics of those countries were - at the time - more or less compatible with those of the Nazis (minus the paranoia), both Brazilian Integralism and Argentine Peronism can be - broadly - described as fascist.
During the war all South American countries maintained a position of neutrality and resisted the pressure from the US to side with the Allies, and most of them only eventually sided with the Allies and declared war against Nazi Germany at the last stages of the war, when trade routes with Germany became unmaintenable. Brazil is the notable exception, the Brazilian Expeditionary Force joined the war in September 1944, still only after intense courting and political and financial pressure from the US and after it became clear that Nazi Germany had little hope of winning the war.
At the conclusion of the war, South America was the ideal hiding place for Nazis. The already established German speaking communities provided safe harbour, through organizations like the ODESSA network, and the general populace was indifferent to the war, as they haven't been particularly affected by it. The various fascist and autocratic regimes of the area couldn't care less about doing the Allies any favours, and at the same time had little reason to antagonize the influential German speaking communities.
At the aftermath of the war and during the first years of the cold war, ties with the US were broken and the relationship of several South American countries with the US became antagonistic. For a relatively short time, the Nazis flourished in South America, especially Argentina, taking full advantage of the fragile political climate, the US - and by association the Allies - were seen with immense distrust. A prime example of the indifference of local authorities towards the Nazis at the time is Josef Mengele's brief detention in Buenos Aires in the late 50s - he was let go, even though his identity was known.
There was one South government that was openly sympathetic to the Nazis DURING World War II, and that was Argentina. This arose from the fact that the country was furthest away from the U.S. and Great Britain, and also had trade friction with both BEFORE the war, while its relations with Germany was decidedly better. Futhermore, Argentina has the largest proportion of Germans (and Italians) of any South American country.
The end of World War II didn't change much. Argentina remained pro-German, anti-Anglo for at least several decades afterward. As late as 1982, Argentina went to war with Britain over the Falklands, with the U.S. supporting Britain. So Argentinians regard people that we might consider "war criminals," as "political refugees."