How dangerous actually were groups like the KKK to average citizens in the South? If I lived in Alabama in 1871, would terrorism pose a significant threat to me?
Terrorism and the threat of it would affect your daily actions, even if you were not physically attacked, and that is the key goal of terrorism. The argument could be made that the American South in the Reconstruction era was subject to a terrorism campaign that worked. For a century after the end of the American Civil War, racially prejudiced laws remained common in the South, and a wave of black Americans who took political office during Reconstruction were swept out by the terrorism that followed.
Eric Foner's works are some of the best around on the topic, and for an introductory audience, I'd recommend Forever Free, a book that was intended to accompany a TV series on the topic, and as such its straightforward, simple writing is a good introduction.
You've heard of the Klu Klux Klan, but you probably haven't heard of groups like the Red Shirts, the White League and the Knights of the White Camilia, that were also present. Violence was widespread, with political assassinations, the destruction of homes and schools, and the targeting of those who attempted to break the mold of the white South.
Intimidation, above all else, was the goal. Even if you were not attacked, you surely would fear attack, and that was the point of the white terrorism. In the Reconstruction era, black Americans were predominantly living in rural areas as agricultural workers. Historically, it's been difficult for those types of workers to organize ─ they're not physically close, and that lack of density means communication is more difficult. Furthermore, the lack of density diluted political influence and meant it was difficult to form freedmen's militias or other protective organizations.
The goal of white terrorism was to intimidate blacks and Republican voters, disrupt the activities of the Freedmen's Bureau and other agencies, and in that, it was ultimately successful.
Based on your question, while /u/The_Alaskan gave you a very good response about the effects of white supremacist terrorism overall, you were really asking about the "average citizen" in the South. About two-thirds of the South was white, about two-thirds worked in a rural occupation, and of those whites, a large majority were Democrats or politically ambivalent.
For about half of Southerners, then, there would have been virtually no danger unless you engaged in specific behavior.
Terrorism in Reconstruction wasn't like terrorism today - there were no large-scale bombings, or attempts to create mass casualties among innocents and bystanders. The terrorism employed by white supremacist groups was targeted, namely at blacks that were believed to be overstepping their bounds, whites who supported the Republican reconstruction governments, and white women suspected of having violated a strict sexual code (and not just by engaging in sex with African-American men).
While the danger to African-Americans (which The Alaskan covered) was substantial, and the danger to Republicans - whether native or not - was high (see: "Chicken" Stephens, an NC Republican murdered by Klansmen during Reconstruction), that violence was of little danger to Southern whites.
What you run into, though, is one of the following: criminality by brigands and former soldiers in the early days of reconstruction being described as "terrorism", which is frankly a misclassification; or the actions of the United States government being described as "terrorism". See, for instance, the Kirk-Holden War. After the end of Reconstruction, historians and writers influenced by White Supremacy tenets would ascribe terroristic motivations to the agents of Reconstruction. To the extent those classifications are politically-motivated myths, modern historians consider them to be false.
SOURCES: Rable's But There Was No Peace: The Role of Violence in the Politics of Reconstruction; Kate Gillin's Shrill Hurrahs: Women, Gender, and Racial Violence in South Carolina, 1865-1900; Mark Bradley's Bluecoats and Tar Heels: Soldiers and Civilians in Reconstruction North Carolina