I understand that alien is used as a synonym of foreigner, as well as a synonym of extraterrestrial.
I suspect that people from other planets wouldn't mind being called aliens. But is it wrong (offensive or somehow inappropriate) to call a person (from Earth!) visiting an English speaking country an alien?
The term alien is actually the technical term used by governments to describe someone who is not a citizen (of either the naturalized or natural born varieties). For example, one of the major immigration forms for the US is entitled "Application for Immigrant Visa and Alien Registration"
I would say it is not at all inappropriate, and any offense would really only be attributed to confusion between the two meanings.
As many people have said, "alien" is a term used by governments to refer to non-citizens. However I'm not sure how often that meaning is used outside of the immigration department. I'm sure for many English-speakers the word's first meaning is extraterrestrials. It certainly is for me.
Using this word sounds "marked" to me. That is, I'd wonder why a person was saying it unless they were a government worker. I'm not sure if there would be cause for offense, however.
Like many terms, it is not offensive in itself, but can be used in a distancing way that others, and so in different contexts it can be neutral, mildly rude, or strikingly offensive.
It is perhaps best considered as cold and technical. There are times when being cold and technical is appropriate, but there are also times when it is better not to be cold and technical in dealing with people's feelings. There are also times when people are pointedly cold and technical.
In the U.S. "alien" used to be the official legal term for non-citizens. However, this is changing. The most important residence document (green card) used to be officially titled Resident Alien Card, but is now changed to Permanent Resident Card. Government bureaucracy is vast, so I'm sure the change some forms still carry the "alien" terminology.
This change implies the term is somehow inappropriate, although I don't see it as offensive.
Most dictionaries put the synonym "foreign" before "extraterrestrial". The latter are a more interesting subject of popular culture (e.g. Hollywood movies), that's why some non-native speakers may get an impression that it is the more common meaning of the word.
I think we non-USAians find it odd, since it's far more common to see expatriats referred to as "expats", "permanent residents", "foreign workers" or "temporary workers" in Asia, at least in the press and in general discussions. As ShreevatsaR points out in the comments below, our governments do refer to foreigners with the same terminology (see links below).
