Boheman was born in Stockholm, the son of Ellen Gustava (née Abramson) and Carl Helmar Boheman. His paternal grandfather was entomologist Carl Henrik Boheman. His nephew was actor Erland Josephson. Boheman studied at the Stockholm University College and graduated in 1918 with a Master of Laws degree.
In 1918, he was appointed attaché to the Swedish foreign mission in Paris, and the following year to London. In 1920, he got a permanent position at the Swedish Ministry for Foreign Affairs, and during the beginning of the 1930s he was Sweden's envoy to Istanbul, Sofia, Athens, Warsaw and Bucharest. In 1938, he was appointed State Secretary for Foreign Affairs and held that position during World War II, up until 1945. During the war he was also Sweden's envoy to Paris, so the Deputy State Secretary for Foreign Affairs, Vilhelm Assarsson, had to step in as Acting Secretary on a number of occasions. He was appointed Ambassador of Sweden to the United Kingdom 1947–48, and Ambassador to the United States 1948–58.
He was a member of the Riksdag 1959–1970 for the Liberal People's Party, the Gothenburg constituency, where he sat in the First Chamber of the then-bicameral Riksdag. He was Speaker of the First Chamber from 1965 until 1970, when the two Chambers merged into one.
Alongside his political mandates Boheman was also chairman of the board of directors of Saab Automobile (1958–1970), Skandinaviska Enskilda Banken and several other companies within the heavy industry sector.
Boheman was married two times. His first marriage was to countess Gunilla Wachtmeister 1919–1927, daughter of university chancellor, count Fredik Wachtmeister and baroness Louise af Uggla, and secondly to Margaret Mattson 1932–1979(his death) daughter of wholesaler Allan Mattsson and Karin Danielsson. Erik Boheman is great-grandfather to actor Richard Ulfsäter. Erik Boheman died on 18 September 1979 in Gränna, Sweden.
He received honorary degrees at Gustavus Adolphus College, Saint Peter's College, Augustana College (Illinois), Tufts College and the Uppsala University.
On 6 June 1968, he received the Royal Order of the Seraphim.
In the Sveriges Television movie, Fyra dagar som skakade Sverige (Four Days that shook Sweden - The Midsummer Crisis 1941), 1988, the role of Boheman is played by Swedish actor Lars-Erik Berenett.