While the policies and actions of the United States do often have a profound affect on other countries around the world, I do not think we should allow non-citizens to vote in any of our elections. I don't even think non-citizens living here should have the right to vote even though I have heard there are people advocating for that.
Can you imagine the magnitude of the expense of conducting an international election every 4 years? Our president is not a dictator, so people around the world participating in electing him alone would not likely have much affect on U.S. international policies.
People in all other countries would have to participate in electing our Congress as well to have a real say in U.S. policies and actions. We don't even allow all the people in this country to participate in state elections and yet the senators and members of the House have a much more immediate affect on those of us living in different states than they do, than they have on the world at large.
I don't want to vote in foreign elections either. The hoopla leading up to our elections is enough. Imagine having to put up with all the wrangling in other countries and all of their campaign ads and debates in addition to our own because we had the right to vote in their elections. No thank you!
So again, the long and the short answer to whether we should allow non-U.S. citizens to vote in any of our elections is no. I do think however, that all U.S. citizens of sound mind should be allowed to vote.