Usually, which verb you use depends on two things: whether you consider the collective noun to be a single unit or to be made up of individuals, and whether you're American or British.
Even though I'm American, I think of 'crew' as being a collective noun (i.e. it uses the singular case to refer to everyone aboard), thus I'd prefer "crew's lives", and adding the "of" just makes the entire sentence weaker, such as using "the survival of the crew".
The actual sentence, if anyone wants to pick the first-draft, uncorrected version, apart, is:
That demonstrates that, rather than being a risk to a crew's lives, allowing your AIs to act independently motivates them to try harder.
Actually, it would probably work just as well with:
That demonstrates that, rather than being a risk to the crew's survival, allowing your AIs to act independently motivates them to try harder
That sounds better, because it removes the competing singular "crew" and the contrasting plural "lives", without the awkward addition of an "of" phrasing.