I would check out Craig Robertson's The Passport in America: The History of a Document (2010). There are many documents that could be considered predecessors to the modern travel visa, but visas and passports are much more the product of modern state development and immigration control. The book mentions the shift from localized forms of communal identity (character witnesses, non-official documents like "safe passage" letters) to a standardized form of identity used by the state to keep accurate records that produced passports.

According to Robertson, the first federal act to specifically utilize immigration visas in the U.S. was the Immigration Act of 1924, which mandated quotas for immigrants from specific regions of the world (it was mostly aimed at immigration from Southern & Eastern Europe and southeast Asia). Chinese immigrants were already denied immigration visas under the Chinese Exclusion Act of 1882 (and the Immigration Act of 1917, which introduced an "Asiatic Barred Zone"), and visas available to other immigrants were capped at two percent of the current U.S. population of that group. In practice, this ended up raising quotas for many western European countries (many long-time residents of the U.S. with English heritage were counted as "British," for example, boosting the quota for that country) while diminishing them for countries like Italy, Poland, Greece, or Russia.