The PR problem
As Arcanist Lupus pointed out, it might not have been a good thing for Hogwarts' image to host a werewolf. There were already a lot of complications to begin with, as Lupin explained.
"But then Dumbledore became Headmaster, and he was sympathetic. He said that as long as we took certain precautions, there was no reason I shouldn't come to school..." Lupin sighed, and looked directly at Harry. "I told you, months ago, that the Whomping Willow was planted the year I came to Hogwarts. The truth is that it was planted because I came to Hogwarts."
Harry Potter and the prisoner of Azkaban, chapter 18, "Moony, Wormtail, Padfoot, and Prongs"
The "human" problem
Dumbledore is depicted as quite caring, and would be - I think - the kind of guy to give second chances on the basis that even if the joke was stupid and deadly, Sirius was still a kid at that point. And as Hitwizard highlighted, he wouldn't want him to become bitter and fall into the dark side. There's a quote about preserving Draco's soul which would indicate that Dumbledore thinks that way.
"That boy’s soul is not yet so damaged," said Dumbledore. "I would not have it ripped apart on my account."
Harry Potter and the Deathly Hallows, chapter 33, "The Prince's Tale"
But the last thing which could be considered (at not the least), is Lupin himself. Dumbledore moved heaven and earth to give Lupin the most normal life possible despite his "furry problem". Expelling Sirius would not only deprive Lupin of 33% of his friends, but might make Lupin feel guilty, even if it wasn't his fault. Lupin already felt guilt for not stepping up to James and Sirius bullying Snape.
Remus functioned as the conscience of this group, but it was an occasionally faulty conscience. He did not approve of their relentless bullying of Severus Snape, but he loved James and Sirius so much, and was so grateful for their acceptance, that he did not always stand up to them as much as he knew he should.
Source : Lupin's page on Pottermore