An associate of Meyer Lansky, Greenbaum joined his organization on New York's Lower East Side in the mid or late 1910s. During Prohibition, Greenbaum began working with the Chicago Outfit managing the southwest division of the Trans-America race wire service in 1928. Sent to Las Vegas shortly after World War II, Greenbaum gained control over syndicate gambling operations, with Morris Rosen and Moe Sedway. With Sedway, Greenbaum ran the El Cortez in 1945 until he was asked by William R. Wilkerson to manage casino operations for the Flamingo Hotel. In 1946, Bugsy Siegel took over construction and creative control of the Flamingo until it was shut down in January 1947 due to mounting casino losses.
After Siegel's murder later in 1947, Greenbaum brought the struggling casino out of debt within several months, eventually controlling several other syndicate casinos and bookmaking operations in Arizona within several years. Greenbaum planned to retire to Arizona and rejected offers to run the Riviera at the behest of Tony Accardo. After his sister-in-law was murdered, he accepted the job. As a leading syndicate leader in Las Vegas, Greenbaum would later order the deaths of Tony Brancato and Tony Tombino for robbing a syndicate hotel. Shortly after becoming manager of the Riviera Hotel, Greenbaum's excessive gambling, womanizing, and drug habits eventually caused him to begin skimming from casino operations. His embezzlement was soon discovered by the Chicago syndicate, and on December 3, 1958, both he and his wife were found in their Phoenix home with their throats slashed.