Though the lines "the albatross / About my neck was hung" (passive voice) don't indicate who hung the bird around the titular mariner's neck, we can infer that the crew of the ship (the other mariners) did so. There are a couple clues.
First, the mariner says, earlier in the poem,
And I had done a hellish thing,
And it would work 'em woe:
For all averred [said], I had killed the bird
That made the breeze to blow.
Ah wretch! said they, the bird to slay,
That made the breeze to blow!
In other words, all the others on board blamed the mariner because he had shot the bird that they thought had brought the favorable winds.
Second, the mariner says, immediately preceding the lines in question,
Ah! well a-day! what evil looks
Had I from old and young!
This reiterates that the other mariners blamed our mariner for their misfortune, and you can infer that this is why, in the next two lines, completing the four-line stanza, and rhyming the 4th line with the 2nd, we get
Instead of the cross the albatross
About my neck was hung.