"Climb down a tree" is perfectly acceptable and I think it's the most common way to state this sentence.
If you use the word "climb" as a verb, one of its meanings is 'to go up'. For example, you could say "The airplane climbed to 10,000 feet."
However, that's only one of its definitions, and it's not the correct definition in the context of "Climb down a tree".
In "Climb down a tree", the word climb refers to the physical action of climbing, moving yourself by using your hands and feet. When using this definition of the word "climb", it is often followed by a preposition that indicates where you are climbing or what location you are climbing to.
Example sentences: "Climb over the wall"; "Climb across the bridge"; "Climb down from there."
If you want a word that specifically means 'to go down', you can use the word 'descend', but this would sound too formal in most situations. You could say "descend the tree" or "descend from the tree", and it would convey the same meaning as the original sentence, but it would sound very weird in everyday conversation.