Answers to this are going to have to depend on politically sensitive judgements, I think, because most of the impact of politicians on existential risk will be indirect and involve things like the overall prosperity of the nation they're leading. Let's look at some classes of existential risk:

• Asteroids and other spaceborne hazards: prefer whichever party will lead to more technological progress in, say, the next 50 years. That will depend on science and education funding (probably prefer the Democrats), on overall national prosperity (prefer whichever party you think will handle the economy better), perhaps in complicated ways on involvement in major wars (maybe too complicated to call even if you think you know which party will lead to fewer wars).
• War (nuclear catastrophe, out-of-control biological warfare): prefer whichever party you think will lead to fewer really big wars in, say, the next 50 years. That's a very political question, and partisans of either party will surely claim that their preferred policies will produce less war.
• Terrorism (ditto): probably not actually a credible existential threat (I'm not even sure war really is).
• Societal collapse: well, what would cause that? Resource exhaustion? (Prefer whichever party will (a) advance technology leading to new resources and/or alternatives and (b) reduce resource consumption if necessary; the latter is probably the Democrats but your opinion on the former will probably match your party affiliation.) Social instability following from huge technology-led unemployment? (If that's coming, probably neither party will help you.) Conflict between social groups? (Political question again. There isn't clear agreement even about, e.g., the sign of the effect of increased immigration.)
• Runaway technology such as AI: probably doesn't have much to do with who's in government. You might prefer whichever party you think will lead to less technological progress, but that leaves you more open to (a) other existential risks and (b) runaway technology developed elsewhere.

I'm seeing scarcely anything here whose answer doesn't depend on things about which people disagree along political lines.

My own answer to your question is: the difference might be quite large but it's very indirect and complicated, so I see rather little prospect of figuring out which way it goes, so I'm going to carry on voting on the basis of things I actually have (or at least fondly imagine I have) some prospect of understanding. I have (or think I have) some ability to predict, on a timescale of a few years, the effect of one party's victory on my own household finances, the risk of some possible near-future wars, the welfare of poor and vulnerable people, the competitiveness of the nation's businesses, etc., and looking at those is probably more effective than trying to guess their very indirect effects on x-risk.