My book-in-progress is historicizing both the concept of the "social problem" and consequently the "social problem film." This has involved an intellectual history of the former and a reception study of the latter. The word-mapping is a good, if very partial, check to see how representative either pursuit is.
The rise in "social problem" usage does at least correspond, roughly to first progressive discourse and second to functionalist sociology. Correlation does not equal causation, but at least it's not contradicting my main argument. The map of the "social problem film" is interesting because it suggests the genre term has become more solidified in the last few decades.
One caveat though: the ngram viewer maps only books, meaning that popular periodical usage is not present. I can attest the term is more prevalent in the 1940s than in the 2000s.
There are some other problems to consider. The search is case sensitive; to use an example from Bady's post, chart "negro problem" and "Negro problem" and you will get two very different timeframes. The chart also does not track tone or context. For that I would recommend the Corpus of Historical American English, which does more or less the same thing Google's service does, but with the context preserved. The COHA gives some further limitations of the Google charts and critique of their accuracy.