"the story begins in year 1937" vs. "the story began in year 1937"

Question

Answer

First, it is not correct to write "year 1937". You could say "in the year 1937" but "1937" is simpler and more common.
Otherwise both sentences are correct.
The present tense "begins' can be used when narrating a story - it is sometimes called the historic present and it has the effect of creating a feeling of immediacy.
Otherwise, use the past tense "began".