It must have been at least twenty years ago I looked up what Earl Browder had to say about the Communist Party's policy toward religion: it was in an issue of The Communist published between 1933 and 1935, I think. While I can't find the appropriate issue on the Internet, a web search suggests that what I have in mind is the report of a 1935 discussion with a group of students at Union Theological Seminary. And there is this publication, which almost certainly reprints the article in question:
Browder, Earl. "Religion and Communism," in Communism in the United States (New York: International Publishers, 1935), chapter 22, pp. 334-349.
This is an entirely different publication:
Browder, Earl. Religion and Communism. New York: Workers Library Publishers, 1939. 16 pp. (Address delivered at the regular morning service of the Community church in Boston on March 5, 1939.) Also available via Scribd.
And there is this:
Browder, Earl. A Message to Catholics. New York: Workers Library Publishers, June 1938. 16 pp.
Programmatic statements do not, of course, give us a full picture of the orientation stated here in practice, especially, given that in the USA as elsewhere, the CPUSA had to ingratiate itself with a variety of religious populations. I have not systematically studied this period, so I will confine myself to a few stray references. The Communist Party of the United States: From the Depression to World War II by Fraser M. Ottanelli depicts a sea-change in the Party's style with the advent of the Popular Front. An anti-communist study, Communism and the Churches by Ralph Lord Roy (Harcourt, Brace, 1960) credits Browder with putting a halt to the chronic ridicule of religion in the Party.
Naturally, this would be a treacherous minefield or tightrope to walk (take your metaphor of choice). I'm no fan of Browder, Stalinism, or pandering, but Browder's bold statements of 1935 should be studied today. In comparison to the flabbiness of the left today, they are exemplary.