Departing passenger lists for that time period are not available. First class passengers were often listed in newspapers of port cities. I found the following in the NY Times for 29 Sep 1931, p. 20. I could only download the article in pdf format, which does not work here, so I retyped the short item.
Passengers for France and England, in the Cunarder Aquitania, include:
Sir Arthur and Lady Dickinson
Col. and Mrs. A. B. Carrington
Capt. H. B. Taylor
Sir Enoch and Lady Hill
Sir Allan and Lady Powell
A. K. Jennings
Mrs. R. W. Van Rensselaer
Mr. and Mrs. H. F. Rice
Dear Ms. Lindsey,
Thank you for posting your request on History Hub!
United States ports were not required to maintain departure passenger lists. As passenger names were occasionally listed in a newspaper for the departure port as traveling on a certain ship on a certain date, we suggest that you search the local newspaper archives for passengers on that voyage. Please check for access at your local library as many library systems subscribe to these sites, making them free for their patrons.
We searched online and located a biography of the RMS Aquitania that may be of interest.
We hope this is helpful. Best of luck with your research!