When the future King Kamehameha IV of Hawaii toured the US as a prince, he wrote in his diary about a bad experience on the train:

"I found he was the conductor, and took me for somebody's servant just because I had a darker skin than he had. Confounded fool;. the first time that I have ever received such treatment, not in England or France or anywhere else........In England an African can pay his fare and sit alongside Queen Victoria. The Americans talk and think a great deal about their liberty, and strangers often find that too many liberties are taken of their comfort just because his hosts are a free people."

This is a fairly regularly asked question that should really be added to the FAQ.

Here is some reading to get you started while you wait for an answer.

How were non-black minorities treated in the Jim Crow South?

What was Jim Crow/segregation era America like for non-black minorities?

I can answer your question about Japanese in the South.

During World War II, there were two internment camps for Japanese Americans in Arkansas and one in Mississippi. In spite of some opposition from state politicians to locating the camps there, the benefits for the federal government (ie. the land was cheap and it was thought that the Japanese might be able to make something of it) were too many to ignore. The opposition was primarily due to the difficulties that their presence would create for the racial dividing lines and because of general racism against the Japanese Americans.

Southern leaders had a difficult choice when dealing with the incoming Japanese Americans. If they labeled them as colored, then it could lead to difficulties if they did work that was traditionally for whites only. This would blur the color line and could lead to blacks believing that such work or opportunities were open to them as well. This was a non-starter for the white southerners. Hence, they went with option two.

Treating the Japanese Americans as white in regards to Jim Crow laws was not without its problems, however. Primary among these was that in no way did white southerners believe that Japanese were their equals. The only reason that they were white in the eyes of the law was to keep as much distance between white and colored as possible.

Hence, to answer your question, Japanese (and Chinese, Italians, Greeks, Spanish, Arab, AND Mexicans) would have sat on the white side.

HOWEVER, this is not to say that there was equal treatment at all. Japanese Americans (both internees and soldiers stationed in Southern bases) faced discrimination when they went into towns. For example, when Private Louis Furushiro went into a cafe while going to visit his sister, who was interned, a man named William Wood heard about it and fired his shotgun at him. In another case, a man "arrested" three internees and shot one of them in the leg. The punishment for these crimes was extremely light because the government did not want to deal with public opinion issues that might arise.

Japanese Americans in Arkansas were also made unwelcome in the state's white universities. They were not allowed to enroll because it might make the entry of African Americans easier. In many cases, the internees could not get jobs for the same reason.

I hope that this adds some to your understanding of the delicate balance that the Jim Crow South attempted to walk and how various ethnic groups could upset that balance.

Source

Ward, Jason Morgan. “‘No Jap Crow’: Japanese Americans Encounter the World War II South.” The Journal of Southern History 73, no. 1 (February 2007), 75-104.

EDIT: grammar

Can I add a further detail to the question asked by OP?

My question is this: would the answer vary depending on the particular Mexican person in question? For example, on whether he/she appeared to be more of Spanish or of Indian descent?