harry potter - Why didn't Dumbledore prevent the prosecution of Sirius Black?

Dumbledore didn't see Sirius as innocent at the time of the incident:

(Dumbledore) 'A street full of eye-witnesses swore they saw Sirius murder Pettigrew. I myself gave evidence to the Ministry that Sirius had been the Potters’ Secret Keeper. Sirius didn't act like an innocent man.’

(Hermione) ‘But you believe us.’

‘Yes, I do,’ said Dumbledore quietly. ‘But I have no power to make other men see the truth, or to overrule the Minister for Magic ...’

Prisoner of Azkaban - page 287 - Bloomsbury - Hermione's Secret

Also see Why didn't Dumbledore insist that Sirius Black be questioned under Veritaserum?

So, mainly, Dumbledore didn't believe in Sirius's innocence until the end of Prisoner of Azkaban. Further, he did not have the power, ostensibly, to override legal decisions made by the Minister for Magic or the Department of Magical Law Enforcement (that said, we've seen Dumbledore step around Fudge more than once. For example, when he creates the Portkey in front of Fudge, which is against regulations.).