The Jim Crow laws were abolished in a democratic way. They were abolished through court decisions and through legislation.
The Jim Crow laws were laws that mandated racial segregation in all sorts of public facilities, particularly in the South, for many decades after Reconstruction. They were held to be constitutional by the Supreme Court of the United States in its decision in Plessy v. Ferguson in 1896.
The Jim Crow laws were largely unchallenged until after World War II. At that point, the first major changes to these laws came through court cases. African American activists brought cases that were aimed at Jim Crow education. They got courts to declare that things like roped-off sections for black students in graduate school were illegal under the 14th Amendment. The Supreme Court said the same about a separate law school for African Americans in Texas. Finally, in 1954, the Brown v. Board of Education decision outlawed segregation in education.
The movement for black rights then turned more towards legislation. The Civil Rights Movement culminated in the passage of the Civil Rights Act of 1964. This made all racial discrimination in public accommodations illegal.
In these ways, the Jim Crow laws were abolished democratically. They were abolished by court cases and by legislation, not by violence.