As it can be seen, the question focuses on the number of the subject and verb, whether it should be single or plural.
Option A: The phrase 'either John or David' presents a choice between two people, 'John' and 'David', out of which one would be performing the action. Hence, the subject is singular.
The verb that follows 'is' is singular and agrees with the subject, which is also singular. The verb and subject must agree with one another in terms of number in a sentence. So, no change is necessary and option A is correct.
Option B: 'Are' is a plural verb, whereas the subject is singular. It does not agree with the sentence.
Options C and D: Similarly, 'are' is a plural verb and does not agree with the singular subject.
Hence options B, C and D are incorrect.