The sit-ins of the time had but one purpose: to challenge segregation and its legitimacy. In that sense, they were also challenging a status quo in the South that had existed for well over a century. The participants were arguing for equality, which went against the majority beliefs of the South - that whites were better than blacks, and that the white hegemony was the proper order of things. It meant that movie theatres, cafes, restaurants, milk bars, and other similar venues would no longer cater only to one race or the other, but to both. This was an encroachment of white society at every level. Mixing with blacks wasn't an option that most people wanted to exercise.
The danger of sit-ins comes from the reaction and reprisals that followed. When sit-ins were staged, they were done with a non-violent intent. However, it was one sided non-violence, and they were at risk of being manhandled out of diners, arrested, beaten, or even having their families harmed at a later point.
Throughout the long Civil Rights movement there are instances of white reprisals against those who challenged the status quo, occurring both before and after the sit-ins. In South Carolina in 1947, a judge made a ruling that black voters could not be prevented from voting in the Democratic primaries. This was followed up with threats of violence, and in 1950, the judge left the state after he and his wife received several more threats. The reasons? He had entertained black guests in his home, and he had made several more rulings which went against the segregation of black and white Charlestonians.
In another case, freedom riders who rode buses from Washington D.C. down to the South in an effort to break segregation were met with violence. Arthur Marwick writes in his book The Sixties about the reception they met in Alabama: "[Outside Anniston] one of the buses was destroyed by fire, and vicious mob attacks took place in Birmingham and Montgomery".
The third example is the Albany Movement. It was founded by members of SNCC and the NAACP, and Dr. King aided the group. They were protesting segregation too, because despite it already having been ordered by authorities, it remained in place. In 1962, King was arrested with several others while protesting, "during an episode in which the police led the way in manifestations of extreme white violence".
I bring up these different events to show that white on black violence was not a rarity, but was in fact a common occurrence in the South at this point. In the face of black protests over segregation and inequality, white populations in the South had reacted with extreme prejudice, committing numerous acts that would be regarded as criminal today. Arson, assault, and police brutality all featured. In the case of the judge in Charleston, intimidation and threats of violence were also used, but came at later dates in the form of recrimination for previous acts.
Knowing all of this, we can see the threat that the sit-in participants were exposed to. In their immediate situation, violence from shop owners, police, or members of the public was a possibility in every protest. As well, they could face later consequences in the form of harassment, refusal of service, and threats of as well as actual violence to their families or homes.
Sources:
Arthur Marwick, The Sixties.