Arthur was born Gladys Georgianna Greene on October 17, 1900, in Plattsburgh, New York, to Protestant parents, Johanna Augusta Nelson (1874–1959) and Hubert Sidney Greene (1863–1944). Her Lutheran maternal grandparents were immigrants from Norway who settled in the American West after the Civil War. Her Congregationalist paternal ancestors immigrated from England to Rhode Island in the 17th century, and in the 1790s, Hubert Greene's great-grandfather, Nathaniel, helped found the town of St. Albans, Vermont. Gladys had three older brothers: Donald Hubert (1890–1967), Robert B. (1892–1955) and Albert Sidney (1894–1926).
Johanna and Hubert had met and married in Billings, Montana. Hubert moved his wife and three sons from Billings to Plattsburgh in 1896 so he could work as a photographer at the Woodward Studios on Clinton Street. The product of a nomadic childhood, Gladys lived at times in Saranac Lake, New York; Jacksonville, Florida, where George Woodward, the owner of Plattsburgh's Woodward Studios, had opened a store; and Schenectady, New York, where relatives from her father's family had settled. The Greenes lived on and off in Westbrook, Maine, from 1908 to 1915 while Hubert worked at Lamson Studios in Portland, Maine, again as a photographer.