A few things to keep in mind. First, the Great Depression happened well into the period of Chinese (Asian, really) Exclusion. In the 1930s the number of Asian Americans was very low compared to whites, blacks, Mexican-Americans and other groups. They were singled out early and effectively, so that's one reason you haven't read much about what was happening; they were a tiny group thanks to effective institutionalised racism of the period.

Leading up to the Great Depression, despite the major restrictions on immigration and the difficulty in entering the country, a number of Asians were still going to the US for the economic opportunities provided. One of the few jobs available to the Chinese — washing clothes — was still a profitable endeavour in the 1920s, and by that period about 30% of all employed Chinese in America were working at laundries.

It cost around $3000 to buy a laundry, and the workers could make around $50 a week, which was plenty of money to send back and support families in China. Families in America were even less common, again because of restrictive immigration, so those that were immigrating (or migrating, more typically) were men coming alone.

That $50 a week was cut in half during the Great Depression, but it wasn't enough to deter people. That was still an adequate amount to send back to support a family, and still more than they would be making back in China.

Quoting from Iris Chang's Chinese in America because it's already concise enough that I won't do a better job of condensing it:

The Great Depression did not affect the [American] Chinatowns of 1930s as badly as the crisis of the 1870s, largely because of the self-sufficiency of these ethnic communities. The knowledge that they could not get easy access to white venture capital had long ago instilled in them certain protective habits, such as frugality, reliance on family connections, and avoidance of frivolous debt. Isolated from white mainstream America, deeply distrustful of white banks, most Chinese businesses had established their own informal credit systems. Aspiring entrepreneurs would borrow money from their own relatives, or partner with other Chinese immigrants to create a bui [會], a pool of capital into which they would make regular deposits and out of which loans would be made at mutually agreed rates of interest.

Still, as with other Americans, Asians had to become more frugal, and they were certainly effected. You've asked about Americans but Asian Canadians faced similar issues. The depression brought unemployment rates to 80% in Vancouver, where most were living at the time. The general unemployment rate in Vancouver was 28%. The increasing strictness of the exclusion acts meant a lot of Asian-owned businesses which catered to other Asians had to shut their doors, since there were not enough customers to keep them in business.

Hope that helps. My answer is more explicitly about Chinese Americans, but Japanese Americans were facing the same struggles and being targeted by the same racist policies, so much of it is still completely applicable.

Let me know if you have follow ups.

Sources:

• Chang, Iris (2004) The Chinese in America A Narrative History. Penguin Mass Market.

• Lee, Erika (2003) At America’s Gates Chinese Immigration During the Exclusion Era, 1882-1943. The University of North Carolina Press.

• Mar, Lisa Rose (2010) Brokering Belonging Chinese in Canada’s Exclusion Era, 1885–1945. Oxford University Press.

• Siu, Paul C. (1987) The Chinese Laundryman: A Study in Social Isolation. Edited by John Kuo Wei Tchen. New York University Press.