This is an old debate in cryptozoology, just renewed on Cryptomundo. My thoughts as posted there:
First, concerning historical reports of unknown-primate bodies: a report of a body is the same as a report of a live animal sighting. It may or may not be accurate, but it doesn't add anything to scientific proof if the body is not around to validate the account.
There are exceptions, but they are rare: the evidence for "Peking Man" isn't in any doubt, but the bones were examined by qualified experts and photographed before they were lost. No one outside cryptozoology seems to have accepted the "sea serpent" Cadborosaurus willsi (described by two qualified scientists, but the type specimen put forth was only a photo of a misplaced carcass) or Grover Krantz's description of a living Gigantopithecus blacki (where he argued that sasquatch footprints were acceptable as a type specimen).
One important case involves a species for which there are some physical remains, but they are too degraded for DNA. Octopus giganteus, described over a century ago by A. Hyatt Verill based on photographs, witness descriptions, and tissue specimens from a stranded body, was resurrected and seemed headed for acceptance before some scientists disputed it on the basis of tests of the surviving tissue samples: I don't think the question is entirely closed, but it's still accurate to say teuthologists as a group have dismissed it.
The proof demanded of sasquatch is the same as for any other animal, a type specimen. It's been established in recent cases that a type specimen does not have to be dead or kept in long-term captivity. Relatively brief observation and filming of a captive specimen and the taking of blood for DNA will do. Of course, if sasquatch does indeed exist, getting a captive specimen is likely to be even harder than getting a body: we're not talking about trapping a bird in a mist-net here.
The question is whether anything short of either shooting or live-trapping a big, dangerous animal is going to suffice. The question is complicated by the hoaxes of photographs and other evidence. A new photo or video, no matter how good, is not going to do it unless backed up by DNA and/or close observation by independent experts.
Unless the animal obliges us by losing a finger in a bear trap or some similarly unlikely circumstance, I think it comes down to this: if the animal can't be captured, killing a specimen might be what it takes to get the animal and its habitat protected. I say that even though I am certain that, if I personally had a bead on a sasquatch, I could not bring myself to pull the trigger. But I would not condemn the person who did if their motivation was the belief this was the only way to get the species protected.