Explanation

“Peculiarly enough” is the clue that indicates that the blanks oppose each other, and best English language playwright is the clue that tells you that the first blank must be positive, so the second blank must be negative. Crowned is the only positive word choice for the first blank.
In the second blank, demonized is much too strong an attitude to direct towards someone for lacking education; dismissed is a more appropriate word.

Crowned, dismissed.
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Peculiarly enough, Shakespeare has been often (i) ______ as the best English language playwright, and often (ii) ______ as a man lacking the education to write those plays.

The word "peculiarly" sets up a contrast between the different ways by which Shakespeare was regarded.

Now he could only be crowned as the best English language playwright, not stigmatized or castigated.

Well, then he could only be dismissed as lacking the education to write those plays. The word demonized is too strong to describe the dismissal of a man as lacking an education. The word deified is a positive word and is rejected.
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