Evidence of Public Domain Status:
At the bottom is the text «"Copyright-Warner Bros. Pictures Inc." All rights reserved»
Under the terms of the U.S. 1909 Copyright Act the artwork (poster) would have had to be renewed 28 years after publication. A work from 1931 would have to have been renewed in 1958 or 1959. There are no relevant hits in the Artwork volumes (which include works of art; reproductions of works of art; scientific and technical drawings; photographic aorks; prints and pictorial illustrations) for "Public Enemy". There's no evidence Warner Bros. renewed their copyright for this material.
Note that it may still be copyrighted in jurisdictions that do not apply the rule of the shorter term for US works (depending on the date of the author's death), such as Canada (70 years p.m.a.), Mainland China (50 years p.m.a., not Hong Kong or Macao), Germany (70 years p.m.a.), Mexico (100 years p.m.a.), Switzerland (70 years p.m.a.), and other countries with individual treaties.