MAGOOSH OE:
Split #1: idiom. The correct idiom is "known for", which choices (B) & (D) have. The rare idiom "known by" is used for a defining characteristic, and it is not appropriate in this context: choices (A) & (C) make this mistake. The construction "known in" is just wrong, and (E) makes this mistake.
Split #2: the movie is known for two things ---(1) "its sexually explicit dialogue", and (2) "the controversy it provoked." These two must be linked by the word "and". Putting those two nouns one after the other, separated only by a comma, suggests an appositive phrase, but that doesn't make sense here --- (2) is not another way of saying (1)! The comma separation is a mistake, and choices (B) & (C) make this mistake.
Choice (D) is the best possible answer.
FAQ: What is the role of "would be" in correct answer choice D?
"would be" is being used to express a hypothetical situation. This movie was originally made in 1953. During that year (1953), its explicit dialogue created a lot of controversy. We can assume most movies in the 1950s did not have sexually explicit language, so "The Moon is Blue" created controversy because of that.
However, the author is saying that the movie "would be" tame--that is, NOT controversial--if it came out today. Because sexually explicit language is common today, if this movie was released in the present, it would be considered a tame PG-13 (suitable for 13 year old children and older) movie by today's standards.