Though the artifacts of pre-Columbian civilization created a stir from the very first European contacts with the New World in the sixteenth century,
it was not until the latter half of the nineteenth century that Western designers, artists, and crafters were inspired to imitate them.
(A) it was not until the latter half of the nineteenth century that Western designers, artists, and crafters were inspired to imitate them -> we have non underlined and unlined parts in PAST tense, which makes better. Let's keep it.
(B) they did not inspire imitations of western designers, artists, and crafters until it was the latter half of the nineteenth century -> Imitation by is better. Incorrect.
(C) not until the latter half of the nineteenth century was it that they have inspired Western designers, artists, and crafters to imitation -> have inspired is incorrect tense.
(D) until the latter half of the nineteenth century Western designers, artists and crafters have not been inspired to imitate -> have not been is incorrect tense.
(E) they were not inspirations of Western designers, artists, and crafters imitations until the latter half of the nineteenth century -> Structure changed as, we have "they were not inspirations of Western designers". Incorrect.
So, I think A.