John J. Giannini, 51, a staff assistant to Sen. Jennings Randolph (D-W.Va.) and a former customer relations officer for the old Post Office Department, died of leukemia Monday at Georgetown University Hospital. He lived in Potomac.

Mr. Giannini, who handled postal and civil service matters for the senator, was a staff member of the old Senate Post Office and Civil Service Committee before joining the senator's staff in 1975. In a statement issued Tuesday, Randolph spoke of Mr. Giannini as "a man who helped people" and whose dedicated public service is appreciated by those he aided.

In 1961, Mr. Giannini joined the Post Office Department, now the Postal Service. He was director of customer relations for the Washington region from 1964 until 1971, when he became a member of the staff of the Senate Post Office and Civil Service Committee.

A native of Follansbee, W.Va., he graduated from West Liberty State College in West Virginia and earned a master's degree in guidance and counseling from West Virginia University in 1959.

He served in the Army from 1953-55.

Survivors include his wife, Joan, and two children, Lisa and John, all of Potomac; two sisters, Lucy Bates of Follansbee, and Lena Rabb, and two brothers, Dr. T.G. and Joseph Giannini, all of Steubenville, Ohio.

The family suggests that expressions of sympathy be in the form of contributions to the D.C. chapter of the American Red Cross or to the Department of Hematology at Georgetown University Hospital.