Remember, independent clauses make sense by themselves. The other two options do not make any sense, therefore, they are dependent clauses. B makes sense so it is an independent clause.
The answer to your question would be that the group of words that is an independet clause is the following one: The dog needs to go for a walk. That is, the correct option would be B.
An independent clause is a group of words that can stand alone as a sentence because it has both a subject and a verb and represents a complete thought.
