(2) The playwright, born over 400 years ago, had become famous all over the world.
With no other context, (2) suggests that the playwright , at some point, had stopped becoming famous. That would be correct if the intent was to say the playwright's fame had propagated everywhere in the past but was not propagating now. The use of the past perfect (had become) would mean the action was completely finished in the past.
(1) The playwright, born over 400 years ago, has become famous all over the world.
(1) means that the action of the playwright becoming famous began in the past, but did not end in the past. The present perfect (has become) means the action began in the past and continued until recently, but not beyond the present.
Which sentence is more correct is a matter for the author to determine based on which time frame is more accurate.