Growing children shouldn't necessarily be handled with excessive care.
Literal
Necessarily growth-stage children [object-を] too fine nerves using handle [should-not-べきでない].
必ずしも...べきでない is the 'not necessarily should' pattern. 細かい神経をつかって means 'using fine-grained attention/being overly fussy.' An opinion about parenting — don't overprotect children.