It's not a ditransitive verb, as that refers to when a verb takes two objects, like *He gave me a sandwich." Instead it could really be doing one of two things:
Like others have suggested, crying could be a participle modifying world, making started a transitive verb with the whole world as its object. This is a rather clunky use of a participle, however, and an odd use of the verb to start. To start normally takes an inanimate noun -- one that represents an idea/thing, and this leads to the other possible meaning of the sentence:
The object of started could instead be crying in gerund form (as a verbal noun). As a result, the sentence would need to become which started the whole world's crying..., making world possessive of the crying. This is certainly the more grammatical way of doing it, although using the participle is fine for informal, song use.