Were they expected to use the "colored" facilities or sit in the back of the bus? Sorry, just a question that interests me, which I've never heard an answer to.
This part is hard to source, but I recall a history professor saying Asians were restricted by labor laws to working only certain jobs such as cooking or laundry in parts of the South. They also would have found moving into certain neighborhoods a challenge. However, the Jim Crow laws were mostly aimed at disenfranchising the sizable African-American population of the South. The Asian population in the South was modest compared to the West, where more laws targeting Asians were passed. The application of Jim Crow laws to Asians would vary from region to region, town to town, in the South depending on the local population's attitude and the local Asian population numbers. Also wording of laws could play a part: signs or laws for "colored" could apply to Asians, but those applying to "negros" would not.
One example of Asians living in the South are Chang and Eng Bunker. They were ethnic Chinese conjoined twins who settled in Mount Airy, North Carolina. Their mixed race Asian and White children fought in the Civil War for the Confederacy. They died prior to Jim Crow law's being enacted, but their descendants seem to have avoided persecution under these laws. Miscegenation laws were passed eventually that applied to Asians, but these were not exclusive to the South and also were not passed in every state.
This is actually a question in our Popular Questions section.
You can read the previous discussions here. I hope these help.
Edit: This one may answer your questions nicely.