Learn Finnish: Miehistöä ei olisi lähetetty kuuhun. - The crew would not have been sent to the moon.

It is made of three parts: the negative word ei, the conditional auxiliary verb olisi (would be), and the passive past participle lähetetty (been sent). Together, ei olisi lähetetty translates to "would not have been sent".

Why is the word for crew, miehistö, in the partitive case (miehistöä) here?

Because the sentence is negative. In Finnish, the object of a negative sentence must always be in the partitive case. Even though the crew is a complete, countable unit, the negative verb ei dictates the case.

If this is a passive sentence, is miehistöä the subject?

Grammatically, no. Finnish passive sentences do not have a grammatical subject. Miehistöä is the object of the action. However, Finnish often moves the object to the beginning of a passive sentence to introduce the topic, which makes it feel similar to an English subject.

Why does the sentence use kuuhun (into the moon) instead of an outer location case like kuulle (onto the moon)?

When talking about traveling to celestial bodies or space, Finnish typically uses the inner location cases. The illative ending -hun is added to kuu, so you literally say you are sending them "into the moon".