What do you mean by open immigration?
In 1916, if you weren't crippled, a lunatic, diseased, a fugitive (which was pretty hard to check back then), of certain Asian ethniciy, and you paid a 50 cent tax they let you in and would naturalize you in fairly short order.
1924 is when we entered the era of any meaningful restrictions on immigration in the US (and then what wasn't repealed in 1965 stopped being enforced).