Taylor, who scored a double century in the drawn first test at University Oval in Dunedin, was on 70 at tea after he was dropped in the slips on nought.
New Zealand captain Brendon McCullum was on 37 at the break with Taylor, who is yet to be dismissed in his three innings in the series and has now scored 303 runs.
The only wicket to fall in the afternoon session was Kane Williamson, who quickly moved from his lunch score of 21 to 45 before he pushed hard at a Tino Best delivery and was easily caught by Darren Sammy at second slip.
Sammy had won the toss for the second successive game and had little hesitation in asking the hosts to bat on a green Basin Reserve wicket.
New Zealand made an awful start with opening batsman Peter Fulton caught by wicketkeeper Denesh Ramdin for six off Sammy while Hamish Rutherford was dismissed in similar fashion for 11 off Best.
Fulton prodded at a Sammy delivery that moved slightly away with the ball going through to Ramdin before Rutherford attempted to guide a short delivery from Best backward of square, only for the ball to flick the edge to Ramdin.
The double-strike left the hosts floundering at 24 for two and they could have been in further danger when Taylor was dropped by Kirk Edwards at second slip with New Zealand's score on 26.
Taylor and Williamson, however, settled the innings and were beginning to look comfortable in their 88-run partnership before the recalled Williamson, who had recovered from a broken thumb, was dismissed.
The 29-year-old Taylor and McCullum, who combined for a 195-run partnership in the first innings of the drawn first test, were beginning to look ominous for the visitors.
West Indies were unchanged from the first test with fast bowler Shannon Gabriel given a second opportunity to prove himself after he finished the match wicketless for 164 runs from his 32.5 overs.
The third and final match of the series is at Seddon Park in Hamilton from December 17.
(Editing by Ian Ransom)
