As almost anything with the Klan, its difficult to say. There is one theory that some of the founding members were brothers of Kuklos Adelphon, a now defunct fraternity that was popular throughout the South. The original Klan itself was formed by 6 former Confederates to alleviate the boredom of life after the Civil War, and there is extensive material documenting the "pranks" they played on each other and on local African-Americans. One example is the water bottle prank, documented throughout the South during Reconstruction and which was so notorious it was replicated in D.W. Griffith's classic Birth of a Nation (if you want to watch the scene its available at https://archive.org/details/dw_griffith_birth_of_a_nation and the scene starts at 1:59:00). Basically, Klansmen would ride up to the house of a local freedman in the dead of night, and demand that they be brought a bucket of water. One of the Klansmen would then proceed to pretend to drink the entire bucket, secretly concealing the water in a pouch under his robes, and saying "thanks, I haven't had a drink since Gettysburg" or something along those lines. Southerners believed that former slaves were incredibly superstitious and gullible, so they thought that these freedmen would genuinely believe they were Confederate ghosts. The truth is that most freedmen were probably too terrified of the consequences of not playing along with the prank. However, there is little to indicate that these pranks evolved into the hazing of college fraternities in later decades. In fact, hazing predates the Klan. Fraternities like the Freemasons, Odd Fellows and others have had long traditions of testing the commitment of their members. It may have devolved into slightly more violent tendencies in certain college fraternities, but the origin of hazing was to test the resolve of initiates and to put on a show to help them bond as brothers. Most fraternities would have to swear terrifying oaths (not a big deal now, but pretty serious in Victorian America) to become a member, while some like the Freemasons lay a blade on your chest or tie ropes around your neck. Its not a specifically Southern (or even American) thing.
If you want to read further about this topic, Mark C. Carnes has an excellent analysis of ritual and fraternity in 19th early 20th century America called *Secret Ritual and Manhood in Victorian America * You can also read the rituals and initiations of various American and European fraternities of the period here http://www.stichtingargus.nl/vrijmetselarij/frame_en.html to get a sense of the nature of the rituals.
First. You're timeline is a little bit off. College fraternities were a thing before and during the civil war. BUT Greek fraternities didn't become a thing till later, however, they rose from these previously existing fraternities.
I think the generally accepted idea is that, yes, there is a strong difference between southern fraternities and... well.. the rest of them. But it's because they have ties to the other side of the war. So while there probably would be differing views between northern and southern fraternities during the heyday of the KKK, these differences between fraternities predates the KKK. Whether or not there was mutual influence would likely depend more on the individual chapter members of each individual chapter of each fraternity. As you can probably imagine, being white in the south during this time would expose you to a radically different set of ideas than a northerner. But that would be true regardless of fraternity affiliation.
So to answer your titular question, it wasn't so much the KKK that influenced southern fraternities, as much as it was the southern culture that shaped southern fraternities differently. It was, after all, the same culture that allowed an organization like the KKK to form and become a not insignificant factor in how politics were run in the south.
But no, the KKK itself did not influence southern fraternities to become so much different than the northern ones, that shift happened before the KKK came around.