The tourists were forced to follow on after they were dismissed for 213 in their first innings, with New Zealand pace pair Tim Southee and Trent Boult doing most of the damage after New Zealand racked up an imposing 609-9 declared.
Although the Caribbean side batted more responsibly in their second innings on a flat University Oval wicket in Dunedin, they face an uphill struggle to avoid another demoralising defeat.
Boult struck early in their second innings when he had Kieran Powell caught at second slip by Southee to leave the West Indies 18 for one.
Kirk Edwards and Darren Bravo steadied the innings, putting on 117 for the second wicket before the leg spin of Ish Sodhi trapped Edwards in front for 59.
At stumps Bravo was not out 72 with Marlon Samuels on 17.
The West Indies arrived in New Zealand talking up their need for a morale-boosting win after being subjected to back-to-back innings defeats by India.
Instead they have struggled badly, failing to take advantage of a green-tinged wicket on the opening day before allowing themselves to be dominated by the New Zealand quicks.
On a wicket that now holds few dangers, they were largely undone in their first innings by a lack of footwork against seaming, full-length, deliveries.
Six of the top seven in the batting order were caught behind the stumps, with Shivnarine Chanderpaul the sole exception.