Possibly the earliest anthologies of science fiction from other languages translated into English for US readers were two slender volumes of Soviet science fiction published by Collier in 1962 with introductions by Isaac Asimov. The actual editors are not identified, but Asimov's name on the covers kept the books in print through several printings over the following decade. Asimov indicates that the stories were selected with American audiences in mind, and he describes how stories in the first volume are technology-dominant in the manner of pre-1950 western sf, while the second volume extends themes into the perfectability of future societies.
Critic Darko Suvin's 1970 hardcover extends the range to socialist countries Poland, Rumania, Czechoslovakia, Bulgaria, and the USSR, and emphasizes the utopian themes that are imagined to derive from their social experiment. A third of the book is taken by five stories by Stanislaw Lem -- his first works, along with the novel Solaris the same year, to appear in English.
In 1977 the US publisher Macmillan launched a "Best of Soviet Science Fiction" series of hardcovers, most with introductions by Theodore Sturgeon, that included the first English-language publications of nine novels (in seven volumes) by Arkady & Boris Strugatsky. The series also included three anthologies, one edited by Vladimir Gakov (in the photo), one uncredited, and a third credited to the Strugatskys.
The final volume included here is much more recent, from a Russian publisher and spanning a century from 1892 to 1992. It includes one story by the Strugatskys, the title slightly differently translated, that appeared in the first 1962 Collier book.
Most reprinted authors: Ilya Varshavsky (11), Vladen Bakhnov, Dmitri Bilenkin, and Arkady Strugatsky (5 each), then Kirill Bulychev and Stanislaw Lem (4 each). Note that the isfdb listings include translators and titles and authors in their original languages.