They're both considered correct. However, there's no reason for the apostrophe. An apostrophe's purpose is to indicate that something has been omitted as in contractions and possessives. I have => I've Newton his => Newton's Nothing's been omitted in "the 1950s" so there's no reason to use an apostrophe there. Using one is considered ok, but it arguably sends the message that you can just insert an apostrophe any old place with not good reason. So I'd avoid using one here, myself.

Actually, the version with the apostrophe used to be the preferred style, on the grounds that “1950” is a number, not a word. (Compare to “in the nineteen-fifties.”) About 30 years ago, the style changed to the version without an apostrophe, based on the arguments mentioned in the other answers. You should use the more modern one, but older publications that use the apostrophe are in no way wrong. (Note: this answer refers to US usage, UK usage may or may not have a similar history. Also note that this is a question of style, not grammar or spelling, and may vary from institution to institution, even in the same region.)

In the 1950s is correct.

B is correct.

B. the 1950s