Although the mariner is the one who shoots the albatross, a friendly bird at first viewed as a good omen by the crew, with his cross-bow, all the sailors share the guilt for the crime. That is because even though they initially berate the mariner for slaying the bird "that made the breeze to blow," before long they change their minds and agree that it was "right ... such birds to slay that bring the fog and mist." The double-minded crewmen change their minds again when the ship enters the doldrums. Eventually some sailors sense in their dreams that the Polar Spirit has followed them from the "land of mist and snow" and is responsible for their perilous state. The men are dying of thirst. Hoping to draw the ire of the Polar Spirit away from themselves and to pin it on the ancient mariner alone, they hang the body of the dead albatross around his neck.
The albatross is hung there "instead of the cross." Whether this means that the mariner was wearing a cross necklace and the others removed it when they hung the albatross there, or whether the albatross was simply hung in a manner reminiscent of a cross necklace is unclear.
The motivations of the other sailors in punishing the mariner this way and their culpability may be hard to discern from the poem itself. To get the best understanding of what Coleridge intended, be sure to read the "glosses" that Coleridge added to the poem in its second edition. Coleridge clearly states in the side notes that the other sailors made "themselves accomplices in the crime" by saying it was right to kill the bird. Later, when their lives are in jeopardy, they "would fain throw the whole guilt on the ancient Mariner: in sign whereof they hang the dead sea bird round his neck." The notes also make it easier to understand the role that the Polar Spirit plays in the drama.
If you don't have a version of the poem that includes the glosses, I highly recommend getting one. They are charming and entertaining in themselves and make the poem unique in British literature.
This is the first poem I taught when I did my student teaching quite a few years ago. Your question has brought back sweet memories.
The albatross is hung around the Mariner's neck because when he kills it, the crew of his ship blames him for the bad luck they suffer afterwards. The winds die down, and they are all stranded at sea. All of the men die except for the Mariner. The albatross is compared to wearing a cross around his neck. It doesn't drop from his neck until he gains the favor of God and is able to pray. He learns that he should love all of God's creatures as he should have loved the albatross.