Were there ever any groups or societies that had a significant number of Caucasian members who were against Anti-Asian Immigration policies back in the days of the old West?
Well, labor unions were almost unanimously against Asian immigration. The International Workers of the World were the sole (major) exception. The IWW was a rather radical trade union, that advocated an anarcho-syndicalist style of workplace management. Eugene V. Debs was one of the founders. So, though it was a trade union and did include a substantial proportion of Caucasians, it was a major exception to the rule in more ways than one.
Probably closer to what your looking for would be found in Protestant missions in the Chinatowns. Of course, there were many churches that were virulently racist at this time, but certain Protestant missionaries did a fine job of reaching out to the Chinese as well. For the sake of time, I am going to quote a paper I wrote for a class earlier this year:
Meanwhile, Protestant missionaries were much less concerned with matters of labor economics, and much more with expansion of their religion’s world congregation. Churches such as the Presbyterians, Baptists, and Methodists, opened church houses in Californian Chinatowns with the purpose of fulfilling “America’s Assimilative Task,” as a 1906 copy of The Wasp labeled it. In addition to the eventual conversion of Chinese-Americans, the missionaries cared about the immigrants on a more human level, and assisted them in achieving a reasonable amount of social well-being (Yu 21). Missionaries often tasked themselves with redeeming and rehabilitating Chinese that had issues with opium addiction, prostitution, or gang violence (Marcus 386). Written works by these missionaries usually conceptualize Chinese immigrant communities as herds of lost sheep—victims of foreign heathenism—who were capable of being fixed by a proper Christian education. They further believed that the Chinese had been brought to the shores of the U.S. for the express purpose of Christianization—an end that the Chinese Exclusion act was preventing the fulfillment of. Because of their noble efforts in combating nativism and assisting immigrants, many Chinese became largely acculturated. While the loss of their home culture is indeed sad, the immigrants who were capable of becoming Americanized were often much better off than those that refused to dispose themselves of traditional Chinese cultural symbols, which left them far less prone to many common arguments made against the immigrant population.
One such missionary was a man named Reverend William Speer. Speer spent many years in Canton on a medical mission, which greatly influenced his understanding of the Chinese people. He reportedly admired their rich culture and vibrant history, and considered them to be the best non-Christian culture in the world. Following his mission in Canton, he moved to San Francisco and opened the first Chinese-oriented Christian church in the city. Speer became an intense advocate for the Chinese-Americans, publishing the first U.S.-based Chinese-language newspaper, The Oriental (or Tung-Ngai San Luk), and regularly advocating against anti-Chinese laws in the California State Congress. He is largely credited with the changing of multiple discriminatory laws (such as the reduction in the cost of the Foreign Miner’s Tax) due to the valiant testimonies he delivered (McClain 547). Speer, while perhaps the most influential of the Protestant missionaries working for Sino-American equality, was only one in a larger movement of Protestant racial reformism. These same moves for equal treatment were made by many Protestant figures, and Protestant churches soon became a location for social refuge.