Hard to say from outside, because most of the foreign policy talk that I've heard so far has been limited to Iraq and Afghanistan. Understandably, sine they are hot conflicts involving US troops, but it's hard to say what Ron Paul or Fred Thompson's policies towards the EU on trade, towards global warming treaties, or towards Chavez-led Venezuela (to pluck a few random issues that probably resonate a lot more outside the USA, and outside this campaign so far, than they do inside).
Generally speaking, which of candidates would be best for America's image in Europe and why?1. It would have to be a Democrat, because mainstram politics in Europe is still, despite some movement away from the political left (e.g. Sarkozy coming to power in France),
way to the left of mainstream politics in the US. No republican candidate is really on the European radar so far - it is still early, mind you - except maybe Giuliani.
2. The biggest signal to the outside world that the USA has changed would be to do elect someone who
looks radically different from GWB. That means Obama and/or Hillary (my own preference would be for President Obama with VP Clinton).
Why Obama? I think he would be a peacemaking president more than a sabre rattler. As well as the obvious physical conflicts, that most of Europe has opposed, Bush has overseen some longstanding arguments with the EU over trade, with green-leaning countries around the world over Kyoto and it's ilk, etc. I think a change of pace to someone who (apparently, and so far) seeks consensus rather than conflict by insticnt would be welcomed, even if the outcome of negotiations didn't always favour the interests of those being negotated with.
But then almost anyone would be flattered by the contrast with What Has Gone Before, at least in mainstream European eyes.
In the Muslim World?Obama. He's nearly one of them - at least that what they might think. They'd be wrong if they thought that would give a free pass, but at least (some of) the Muslim world's leaders might feel able to make a case with words rather than bombers. It remains to be seen how great a change that would be, at either end, however. I doubt any US President will ever say "no, you were right all along. Sorry, our bad." And there will still be maniacs for whom nothing short of the conversion or execution of every
kuffir will be enough.
I hope whoever does get into power uses that power more even handedly - I don't want another US president (or another UK government) that ignores the nationality of most of the murderers of 3,000+ civilians on US soil (or that cans a bribery investigation) because he's golf buddies with one of the princes (or because a factory in a marginal seat might have to close if the people being investigated throw their rattles out of them prams).
In Africa? Obama again, and again because he's "nearly one of them". This time, though, there'd be more reason to think this way. I reckon that during the next US presidential term, the USA will sit up and pay attention to the huge amount of influence (and in return, first-comer resource advantages) the Chinese are gaining all over the continent by the simple expedient of giving unconditional aid and investment, rather than being all neocolonial and saying "you can have X if you agreed to open your markets to our businesses" or "you can have Y as long as you don't mention condoms or other birth control to your AIDS-riddled and meteorically-rising population, because God wouldn't approve".
Asia? As far as most of East and South Asia goes, I don't think they'll mind too much as long as no new trade barriers get thrown up by the new incumbent. The Russians won't mind too much as long as whoever gets in is a little more conciliatory and either backs off on missile defence or extends the umbrella to include Russia (I think the latter would be the most workable solution - even though I don't think the system itself is anything more than a white elephant).
Latin America?If whoever gets into power breaks the habit of the past century or more and stops treating the continent like the USA's "backyard" and starts treating them as sovereign states with their own rights and privileges (and responsibilities), I don't think they'll much care. If you end up electing someone who won't rock the commercial and foreign policy boat (i.e. almost anyone on the candidate list) then South America's attitudes won't change very much.
How important is it to you that our president possesses knowledge and understanding of how to improve our popularity with cultures beyond our borders, and consequently, who then are you supporting?Some visible and expressed knowledge and understanding of the world outside US borders would be a good start and would, on it's own, make the US vastly more popular than it has been under Bush; Clinton (Bill) was very popular outside the USA, especially in Europe. Again, on this score, Obama and Clinton (Hillary) are the only candidates
to score.
For my money, and in my lifetime, the only Republican who'd stand a cat-in-hell's chance of
enhancing America's reputation in the wider world would be Schwarzenegger, and we all know he'd need a constitutional amendment before he could ever stand.
Are we mistaken to value our country's popularity abroad?Not at all. America needs to buy stuff from outside, and sell stuff to outside - isolationism wasn't a workable solution in the 18th and 19th centuries, and in the 21st it is a self-indulgent pipedream. Sometimes there will be points of principle where the immediate self-interests of the USA (or any other nation) should be subsumed for the greater good. I'm not calling for removal of soveriegn powers, here - it has to be governments, ideally elected ones, that make those decisions.
Just for a little
enlightened self-interest now and again, that realises that sometimes the benefits may not be realised within a single electoral cycle, or may only store up a bank of goodwill to be drawn on in case of emergency. (Good luck on that one, though - I don't know of many governments anywhere on the planet that are fully committed to that idea!)
All of that said, however, you would be mistaken if you put
too much value on external popularity - the State Department is not the only Department of State.
EDITED TO ADD:
I started posting and took a long time doing it, so missed
aevans176. I have to say that all of his points are valid too. It just isn't possible to say "Europe thinks such-and-such" or "South America wants America to do so-and-so" any more than it is to say "America thinks ..."
More than anything, though - and I speak on behalf of the entire non-American world here

- we want America to elect someone who
does do nuance.