By the SRD description, an Ape is a Beast, with a Climb speed of 30. However, if somoene is polymorphed into an Ape, by the wording of Polymorph, that Ape cannot climb, because climbing (for an Ape) requires the use of its hands. However, it also says that the statistics of the chosen beast are used, which includes its climbing movement type.
So by RAW, is it the case that someone polymorphed into an Ape cannot climb?
The new form can be any beast ... The target's game statistics, including mental ability scores, are replaced by the statistics of the chosen beast. It retains its alignment and personality ... The creature is limited in the actions it can perform by the nature of its new form and it can't speak, cast spells, or take any other action that requires hands or speech.
ETA: Apes and Baboons, Gorillas, Monkeys, etc DO have hands, not 'paws' (Regardless of what famous short story titles may say)
Also adding that even though the stat blocks for Apes and Baboons DO say they have a Climb speed, those are general, and the spell is specific, which means the spell trumps the stat block.