The answer to this question falls roughly into two categories.
First, there was an element within American society who viewed such imperial adventures as both morally wrong and antithetical to the American ideals of self-government and freedom. This element found its focal point in the Anti-Imperialist League, which was spearheaded by such luminaries as Jane Addams, William James, John Dewey, Andrew Carnegie, and, perhaps most prominently, Mark Twain, who was so effective a critic of American Imperialism that, years after the Spanish American War, Theodore Roosevelt refused to give a talk when he knew Mark Twain would be at the same event.
Consideration of the ideals of the Anti-Imperialist League brings up an ancient question: are the democratic institutions of a republic or a democracy threatened when the same country becomes an imperial colonizer to other peoples? Was it right for a democracy born of revolution to obstruct another people who wanted to throw off their own imperial shackles? Mark Twain certainly didn't think so, and many others agreed with him.
The other considerations were driven by considerations of an economic and racial nature, with the language of the two often mixed together. Members of the labor movement, including Samuel Gompers, had concerns that cheap labor from the Philippines and East Asia would flood the country, diluting the bargaining power of American-born labor.
While racism remains a part of American society to this day, the United States of the 1890s, only thirty-five years removed from the Civil War, was another thing altogether. In a country where the Irish and Italians were considered racially undesirable, and Chinese and Japanese immigration was halted completely or severely restricted, many Americans desired for the United States to be as white as possible and did not want to allow Filipinos into the country.