The first construction, “Falling oil prices have hurt the economies of Gulf countries,” is grammatically correct based on the subject-verb agreement rule that the form of the verb must agree with the number (whether singular or plural) of the subject or of the doer (agent) doing the action. The subject here is clearly the noun phrase “falling oil prices,” which is plural both grammatically and notionally. It thus needs the plural present-perfect form “have hurt” of the verb.
Based on the same analysis, the sentence “Falling oil prices has hurt the economies of Gulf countries” is faulty both grammatically and notionally. In this construction, it’s really a stretch—and a wrong one at that—to consider the phrase “falling oil prices” a gerund, which would make it conceptually singular as a subject.
If you found that this wrong construction is common in the Internet, it’s most likely because some writers mistake it for the correct construction “The falling of oil prices has hurt the economies of Gulf countries,” where the subject is the phrase “the falling of oil prices.” This time, what we have is a gerund phrase that’s both notionally and grammatically singular. It is for this particular construction that the correct form of the present-perfect verb is the singular “has hurt.”
