The most appropriate prop on the set of "Brighton Beach Memoirs" to indicate that the play takes place in 1937 would be a black-and-white photograph of the New York Yankees.
This is the most logical choice as the others would not place a time period or could either indicate something was older or more recent than it actually was. A pitcher filled with lemonade and four empty glass would not give us any indication as to when the play takes place. A color television could be placed as a prop in the present day. Reversely, a wood-burning stove was used long before 1937 and could make the play seem like it was set further in the past than it really was.
