Taylor was on 89 with BJ Watling not out 14 after New Zealand lost two further wickets in the session in their response to the West Indies first innings 367.
The West Indies persisted with the spin attack for an additional 18 overs through to the break after the new ball was due but the rewards were few.
New Zealand resumed the day at 156 for three and apart from Brendon McCullum (12) who never looked comfortable, Taylor, Anderson (39) and Watling moved the score steadily along.
McCullum took seven overs before adding one to his overnight total and showed frustration by slashing at a sharply turning offbreak from Sunil Narine with the safe hands of Darren Sammy clasping the catch in the slips.
Taylor had done his best to protect his skipper through 43 deliveries in the morning during which McCullum was only required to face two Narine deliveries, the second of which was his downfall.
All-rounder Anderson, playing only his fifth Test, required no protection from Narine and was happy to front up with an aggressive approach.
The left-hander took 13 off one over as he tried to hit Narine out of the attack, but he was undone in the next over when he skied a Veerasammy Permaul delivery and was caught by Narsingh Deonarine at deep square leg.
Taylor, with a double ton in the first Test and a hundred in the second, moved steadily towards what would be his 11th Test century.
For the West Indies, Narine has three for 81.