The rest of the West Indies players were fined 30 percent of their match fee from the innings and 67-run loss in the first Test.
At stumps on day three, the Caribbean side were 214 for two in their second innings but still some 172 runs adrift of avoiding an innings defeat and going one down in the two-match series.Brathwaite was unbeaten on 79 and partnered by Shai Hope on 24, with the pair having so far posted 48 in an unbroken third wicket stand.
Taylor, now New Zealand's third most prolific Test batsman of all time, was singled out by the West Indies at the start of the day as their most wanted wicket.
It was the first wicket of the innings for man-of-the-match Wagner and he followed with the dismissal of Jason Holder (seven) to finish with nine for the match after his seven for 39 in the first innings.
Matt Henry finished with 3-57, Colin de Grandhomme 2-40 and Mitchell Santner 1-25.
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The wicket was no longer offering the bowlers any assistance but the fight had gone out of the West Indies, with the downward slide gathering pace from the first ball after the interval when Sunil Ambris was out.
However, just as the West Indies had entere cruise control, Henry provided the breakthrough for New Zealand again as he removed Hetmyer for a career-best 66, which came off 89 balls and included eight boundaries and two sixes.
West Indies then suffered two key blows when Hope was caught at gully by captain Kane Williamson off Boult-failing to keep down one that lifted-and Chase chopped on a cut at a wide ball from Henry to depart for 18.
The right-hander faced 180 balls and struck 13 fours and a six all told as the Black Caps, resuming on 447 for nine, added a further 73 before declaring on 520 for nine, 20 minutes before lunch. Centuries from Collin de Grandhomme and debutant Tom Blundell and Ross Taylor's 93 made sure New Zealand secure a big first innings lead.
The second test starts in Hamilton on Saturday.