Should I list campaign consulting on my CV? (the campaign I consulted for was Republican)
If the experience is related to your academic work then yes, you should list it. Ideally, the party you worked for shouldn't matter, though our politics in the US is very polarized. So, while I likely disagree with your policy stances, should that affect how I view you professionally? 

No, it should not, and many would fight against the tendency to view you negatively, but not all. And while many might not consider it disqualifying, some might consider it a negative factor. Of course this is all true if you had worked for a different party as well. 

There is no need to specify which party you worked for, though it may be obvious from any letters of recommendation. Be honest if asked, of course. 

On the other hand, if you are viewed as very political then polarization would be a bigger factor. 



Two additional thoughts came to mind a bit late. 

First, if you are volunteering, rather than a paid consultant, there would be no need to mention it as it is more part of your private, than professional, life. 

But in general, it is almost always an error to hide anything from your past. People don't normally react well when the learn the truth.