What with the Olympics being in 北京, reporters are pronouncing it in various ways and the question of how to pronounce it is in the news again. Our local paper has an AP article by David Bauder which, Google reveals, is being carried all over the place. Here's one version.
The article is right about the correct pronounciation: [beʤɪŋ] (using the voiced symbols for what are, strictly speaking, voiceless unaspirated consonants); bayjing would be a pretty good English folk spelling. It also gives the correct explanation for the common mispronounciation [bejʒɪŋ], where the <j> is pronounced like the <s> of measure, namely that the sound [ʒ] is somewhat exotic in English, which I explained here four years ago. The article mistakenly asserts that the sound [ʒ] does not occur in English. It is indeed found in English, not only in measure but in such words as azure, pleasure, leisure, and treasure. What is true is that all of the words in which it occurs are loans from French, so the sound apparently has an exotic flavor even though it has existed in English for centuries.
The AP article also addresses the related question of why Beijing used to be called Peking. Although I explained this in some detail in this post, the AP article doesn't get this quite right. It explains that:
It officially changed in 1949, when the new Communist government adopted the pinyin transliteration method for proper names…
This oversimplifies the matter and falsely suggests that Peking and Beijing are different renderings of the same Chinese form in different systems of transliteration, which is not the case: they are different renderings of different Chinese forms. Moreover, the chronology is wrong. pinyin was first promulgated within China in 1958; it became an international standard (ISO-7098:1991) in 1979.