As with other similar devices, alliteration is mostly used in the first and second stanzas of "Journey of the Magi." In the first line, which is part of the opening allusion to a 17th-century sermon by British baptist Lancelot Andrewes, the alliterating /c/ sounds of "cold coming" send a shiver through the line. There is more alliteration in the same quotation, found in lines 4 and 5:
The ways deep and the weather sharp,
The very dead of winter.'
The alliteration here links the words together; collectively, they emphasize the inescapable difficulty of the journey.
In lines 9 and 10 use /s/ alliteration (also known as sibilance):
The summer palaces on slopes, the terraces,
And the silken girls bringing sherbet.
These /s/ sounds link with other /s/ consonance in the same lines ("palaces," "terraces") to convey the luxuries and indulgences that the magi are used to. In their old lives—pre-Christ—they held positions of power, and enjoyed the pleasures that their status brought. The same sound is used to very different effect in lines 14 and 15 (though it's not alliteration—so this is covered in the Consonance section).
In line 18 and 19, more alliterating /s/ sounds combine:
Sleeping in snatches,
With the voices singing in our ears, saying
These /s/ sounds work in two ways. They sound a bit like snoring, highlighting the magi's sleep-deprived state. There's also something sinister about them, like a snake's hiss, suggesting both the hostility encountered by the magi and the voices of doubt ringing in their ears.
In line 27, the two prominent /d/ sounds in "door dicing" evoke the sound of hands, coins, or dice being banged upon a table. Again, this helps build a sense of the general aggression and animosity that the magi found along their way.
Alliteration occurs less frequently in the final stanza, which is much more prose-like in tone as the speaker switches from describing his epic journey to contemplating what the meaning of that journey really is. In this stanza, alliteration occurs mostly on /b/ and /d/ sounds, underscoring the speaker's many doubts about the Birth and Death that he's contemplating. The "Birth" of Jesus turns out to be "bitter" for the magi, and the alliteration highlights this surprising and troubling conclusion.