Point of clarification: In some incarnations (including - and maybe especially - the period you reference), the KKK was targeting some populations we'd think of as "white" today.

In the 1920s, they were standing against "Kike, Koon and Katholic", especially in the Pacific Northwest. The "second-wave KKK" was strongly concerned with immigrants as a threat to white, Protestant society.

The second incarnation of the KKK was started in the South but was popular in the Midwest, Northeast, and West Coast. It was particularly powerful in the Midwest, where it controlled a number of prominent politicians. The first two important figures to take stands against the KKK were both southern politicians, Governor "Big John" Parker of Louisiana and Senator Leroy Percy of Mississippi. Parker asked the FBI for assistance in taking down KKK operations in North Louisiana in 1922 and (also in 1922) Senator Percy walked into a KKK recruiting rally in his hometown and directly confronted the KKK organizers, attacking them in a speech and convincing the town to vote on an anti klan resolution. Parker and Percy received national attention for their stands. Percy's stand was one of principal, he was no longer in the senate and interested mainly in keeping the klan out of his part of Mississippi. Parker was more realpolitik, the klan had murdered 2 people in Louisiana and he was not able to seek justice through ordinary means because the KKK had too much power in the area the murders had occurred.

Politically Percy was a Democrat, but advocated policies offering more equality and educational opportunities for blacks. He was elected to the Senate by the state legislature over a white supremacist candidate who later defeated him in an open election. Parker was also a Democrat at the time of his election as governor, but had been named the Vice Presidential candidate of the Progressive Party in 1916.

These are not every day common man stories, but they serve to illustrate a couple of points. First, the South was controlled by the Democratic Party but not to the exclusion of all others. Republicans were still persona non grata in the 20's, mainly as a response to "Reconstruction" era policies, but the Progressive Party was able to get a hold in some areas as well (Parker ran for governor as a Progressive and got 37% of the vote, not enough to win, but this wasn't a one party system like Communism).

Also, the KKK was never quite as powerful in the South as it was in the Midwest in the 20's. The mythology of American race would lead you to believe KKK is all Southern and all Southerners were pro KKK but neither was the case in the 20th century. In the South the KKK was openly despised by many, and both Parker and Percy found a lot of support for their actions when they fought back against them.

It's a pop history by an amateur historian (one that won the Francis Parkman prize, if that means anything) but the best description of the Percy stand against the klan I have read is in Rising Tide by John Barry. This book focuses on the flood of 1927, the greatest natural disaster in American History, but it also provides a great amount of information on race relations during the 1920's, including the power of the KKK and Barry's theories on why Black voters shifted from die hard Republican loyalists to die hard Democrats.

This is a complex question because the "South" is a very diverse region and the Klan behaved differently in each state. Furthermore, there are plenty of cases of violence that may have not been attributed to the Klan. For instance, the Klan in Alabama was heavily involved with the police, and it is believed that several attacks were carried out in the 1920s and 30s by Klansmen but which were never revealed. (Glenn Alan Feldman covers the issue of the especially violent Alabama Klan in his book "Politics, Society and the Klan in Alabama, 1915-1949). In other parts of the South, like Tennessee or Virginia, it was relatively safer to stand up against the Klan

It also depends on who you are. Julian Harris, the editor of the anti-Klan newspaper the Columbus Enquirer-Sun of Georgia was an important force in the anti-Klan movement and denounced them daily in his paper. His town had about 400-500 Klansmen, and he received threats of arson, murder and was told to leave town. His newspaper press was also sabotaged more than once. One threatening letter he received after communicating with the anti-Klan newspaper the New York World read "DEAR SIR: LEAVE NEW YORK NEWSPAPERS, TROUBLES WITH KKK ALONE. DEAD MEN TELL NO TALES". It is believed that a Klansman was monitoring his telegrams at the local Western Union. Harris and the Enquirer-Sun would go on to win the 1926 Pulitzer Prize for their efforts fighting the KKK (source William F. Mugleston, "Julian Harris, the Georgia Press, and the Ku Klux Klan" Georgia Historical Quarterly, Vol. 59, No.3 (Fall 1975))

Plenty of politicians in the South also made a career fighting the Klan, as the order was not as powerful as most people imagine. The Texas Klan had been notoriously violent and powerful in the early 20s, and Texans were fed up with them. In the 1924 gubernatorial election, Miriam 'Ma' Ferguson won the race on her explicitly anti-Klan platform.

The Klan was actually more powerful and entrenched in certain Northern states than in some Southern ones. To give you an idea, Shawn Lay in Invisible Empire in the West, claims that there were more Klansmen in Indianapolis than there were in the Carolinas. Indiana was the heart of the Invisible Empire, and it was here that it was probably the most dangerous to stand against the KKK (not just in terms of the threat of violence, but also of commercial boycotts and social ostracization)