During the late 19th and early 20th century, the earliest forms of what we would now recognize as "ethnic cuisine" in America started to emerge. For example, starting in the late 19th century in San Antonio, chili was a popular local delicacy which we know was enjoyed by white residents and visitors. In fact, in much the same way that tourists flock to notable outposts selling regional cuisine (when you go to Philadelphia you must get a cheese steak from Geno's, in New Orleans you gotta try the beignets at Cafe Du Monde, etc) today, tourists to San Antonio were trying and commenting on the local chili scene.

Similarly, around the same time, Chinese cuisine was starting to occupy a similar place in cities like San Francisco and New York which had sizable Chinese enclaves. At first, the restaurants were established for Chinese patrons, and the food was as close as possible to what people would have missed from back home. However, soon adventurous whites started to experiment with eating Chinese food. From there, it didn't take long for enterprising Chinese-Americans to open restaurants more focused on white American tastes. The difficulty of obtaining visas to emigrate from China also plays a role here, as it was easier for business owners to get visas.

Italian-American cuisine also started to take shape around this same time, and in similar ways, though it didn't really go full force until American GIs came home from WW2 having grown familiar with Italian cuisine.

However, it should be noted that in both of these cases, the food in question wasn't treated the same way that 2018 American culture tends to treat ethnic cuisines. With the possible exception of a few Chinese restaurants in major cities (which did specialize in affordable fare catered to a mainstream white clientele very early on), it wasn't like "What do you want for dinner, Chinese or Mexican?" but more of an exotic experience or a "when in Rome" mentality of wanting to try something unusual for the adventure of it.

It also has to be said that, for the most part, we're talking about American adaptations of the cuisines in question. Chili is the beginnings of "Mexican" ethnic cuisine hitting the American mainstream, but it's not Mexican as in imported from the sovereign nation of Mexico. It was developed over time by Latin-American settlers in and around San Antonio. Likewise, most of the dishes that were popular on Chinese restaurant menus in the 19th and early 20th centuries bear no real resemblance to 21st century regional Chinese cuisine. Chinese restaurant owners, especially, catered to white tastes early and often and developed a cuisine of their own that has evolved along its own trajectory. Like the connection between Mexican-American food and San Antonio, there's also a strong connection between Chinese-American food and San Francisco. For example fortune cookies are actually descended from a sweet brought to San Francisco by Japanese immigrants and are not Chinese at all.

Edit: In terms of how it was looked upon by white people at the time, it most likely would not have been denigrated so much as seen as an exotic thing to try every once in a while, as an experience. All the while absolutely looking down upon home cooking made by immigrants for immigrants, as eaten by non-white people.

Sources:

Taco USA: How Mexican Food Conquered America, by Gustavo Arellano

The Fortune Cookie Chronicles and The Search For General Tso, by Jennifer 8 Lee.