I am having difficulty deciding which of the following is correct:
- Politics both fascinates and repulses him.
- Politics both fascinate and repulse him.
I think, in part, this is because I am unsure if 'politics' is the subject.
That depends on what you want to say. If you are talking about politics in general, use the singular:
Politics both fascinates and repulses him.
If, however, you are referring to specific politics, for example those of a person, use the plural:
John's politics both fascinate and repulse him.
This is because politics can mean different things. The ones we care about here are (taken from the online Merriam-Webster):
1a : the art or science of government
4: the political opinions or sympathies of a person
So, the first example is using sense 1a and the second is using sense 4.