That's a very interesting question. Don't trust my answer unless other people agree with mine.
The correct answer--I'm _almost_ sure!--is:
"The protection and development of art have important implications for cultural diversity."
Grammatically, "protection" and "development" are both singular nouns, so "protection and development" is plural.
But as a lifelong native it feels strange to me. Even though logically the subject is "protection and development," by the time I've said "development of art" I feel a strong pull to the word "has." I think that in speaking aloud it would be very common to say "has" and that most listeners would not notice any problem.
Quite seriously, if I were writing this myself, I would probably stop and say "I don't really want to solve this problem," and I would reword it, perhaps as
"Protecting and developing art are important in sustaining cultural diversity."
In this sentence, "are" is correct and "is" is wrong. The odd thing is that I feel sure about this, whereas in your sentence, I know what's right but I don't feel sure.
I have two other comments. "Important implications to X" is not right; it should be "important implications for X."
However, there are better word choices than "implications." "Implications" usually had to do with logical connections and thinking. "The new rules have implications for the way we do business. If I understand section 5, it means we will now need to apply for a permit." It doesn't mean cause-and-effect. "A implies B" doesn't mean "A causes B" or "A helps B" or "A nurtures B."
I might rewrite the sentence: "Protection and development of art are important for sustaining cultural diversity." ("Sustaining," "maintaining," "supporting," "nurturing," "preserving" are all possible choices here).