It all depends on where and when you are in the Jim Crow South, an Asian-American may be seated in the colored section, in the white section, or not at all. Depending on if he's in South Carolina, Texas or Mississippi. From a previous answer I wrote:
Asian American's faced the same kind of discrimination as African-Americans during the Jim Crow period. Legislation barred them from intermarrying with whites, attending white schools, they faced social segregation as well as extra legal violence such as lynching. But, the status of Mexican and Asian American's changed depending on their numbers as well as where they lived. The group that has become known as "The Mississippi Chinese" or "Delta Chinese" were a community of Asian Americans that lived in the heart of Jim Crow country in the Mississippi Delta. Arriving during the Reconstruction era, they were subject to racial discrimination but managed to find themselves a niche position in operating family grocery stores. They found themselves operating as middlemen, a third group in a biracial society. As opposed to the Mississippi Chinese, who were stuck in a racial "in-between" neither white nor black. Some of the Chinese of South Carolina (few though they were) were able to successfully integrate into white, middle class society. Attending white schools, marrying into white families and even anglicizing their names from Sing to Sanger. The Citadel, South Carolina's military academy wouldn't admit an African American student until 1966, while ethnically Chinese cadets entered as early as the 1920s. Others would attend the Avery Institute or Catholic school, local private schools for Middle-Upper class African-Americans. Of ten Chinese families in Charleston between 1880 and 1920, five of the wives would be classified as white, three as mulatto, and two as Chinese, demonstrating that White-Asian and Black-Asian marriages were stigmatized to a lesser extent then the somewhat more "traditional" interracial coupling of white and black. The Jim Crow society was able to adapt to these outsiders by identifying the first generation Chinese as White, but their children's race depended on the mother (as Southern American society is Matrilineal, a practice that extends back to slavery.) In regards to legislation, Asian American's had a much tougher time. The 1923 Case U.S. v. Bhagat Singh Thind[1] rejected attempts by South East Asians to gain citizenship by stating " [if the] applicant is a white person within the meaning of this section he is entitled to naturalization; otherwise not." meaning that any non-white foreign person who attempted to gain citizenship would be denied. Another important case regarding Asian American race was the 1927 case GONG LUM v. RICE, (1927)[2] which legislated that because the Asian American community in a Mississippi township did not have a separate school system, they would have to attend the colored school. Thus again, forcing them into a non-white, segregated status. In areas with larger Asian American communities, such as California or Washington, Asian American communities were granted their own separate school system. California had a law that was not struck down until 1947 that forced children of Japanese, Chinese, and Mongolian descent into their own segregated school district.
Sources and Further Reading
By far, the best introductory text on this topic is White By Law by Ian Haney Lopez. It covers the legal construction of race in the United States and is a great starting point into how race, culture and law work and worked in our history.
Asian Americans
Ancheta, Angelo. Race, Rights, And the Asian American Experience, 2nd edition
Habal, Estella. San Francisco's International Hotel: Mobilizing the Filipino American Community in the Anti-Eviction Movement
Friday, Chris. Organizing Asian American Labor: The Pacific Coast Canned-Salmon Industry, 1870-1942
Pei-te Lien, M. Margaret Conway, Janelle Wong The Politics of Asian Americans: Diversity and Community
Nemoto, Kumiko Racing Romance: Love, Power, and Desire Among Asian American White Couples
O'Brien, Eileen. The Racial Middle: Latinos and Asian Americans Living Beyond the Racial Divide
Odo, Franklin No Sword to Bury: Japanese Americans in Hawai'i during World War II