Neither the Western Hemispheres nor the Eastern Hemisphere ARE immune to flooding. The author refers to more than one hemisphere so 'is' (singular) is not appropriate.
The sentence that has the correct subject- verb agreement is the following one: Neither the Western Hemisphere nor the Eastern Hemisphere are immune to flooding.
The verb "to be" can't be singular because the writer mentions the two hemispheres (the Western Hemisphere and the Eastern one). That is why the first option is the correct one.
