From Redditor u/murderhorny:
One question that always bugged me is why didn't Dumbledore help Sirius? I know that he didn't know about the last-minute switch between Peter and Sirius, but he knew that betraying his best friend isn't like him, and Dumbledore is a skilled legilimens, as showed when he read Creature and Morfino Gaunt's minds, he could have read Sirius mind, found out that he is innocent and his word would be enough to free him since his word was enough to clean Snape's name, the headmaster also has a lot of influence in the wizarding politics, this would explain why Sirius was sent to Azkaban without a trial, so that he had no chance of trying to prove his innocence since Death Eaters that have committed equally horrible crimes were given a trial. But why would Dumbledore go out of his way to ensure that one of his loyal order members would be thrown in Azkaban? Because If Sirius was cleaned, he would have taken legal guardianship of the little Harry, and ‘The Boy who lived’ needed to go to the Dursley's because Petunia was a blood relative of Lily, so as long as Harry called the Dursley's house ‘home’ the protection of Lily would live on because of Petunia, so he would spend his formative years in a harsh, but safe, environment. Dumbledore is a man who does things thinking about the Wizarding World as a whole, not in individuals, and on that moment, the best for the world would be to Harry to stay safe at the Dursley's until he was old enough to fulfill the prophecy, and although he didn't dislike Sirius, he needed to put him out of the picture to his plan to work.