The Black Caps began their response to the tourists' 367 immediately after lunch, with Shivnarine Chanderpaul left stranded on 122 when Tino Best was the last man out on the stroke of the interval in Hamilton.
And they found it tough going as they lost openers Hamish Rutherford and Peter Fulton in the afternoon, before Williamson (58) and Taylor (56no) launched a rescue mission after tea, finishing the day 211 runs behind.
Rutherford went when his attempted straight drive off Darren Sammy went straight back to the bowler to leave the hosts 18 for one.
Fulton dropped anchor, scoring 11 off 71 before he was excellently caught by Sammy at leg slip off the bowling of Sunil Narine.
Williamson and Taylor negotiated the rest of the session without incident though runs remained hard to come by.
Taylor attempted to up the rate after tea, highlighted by a couple of boundaries off one Best over, but Williamson was still playing at a crawl.
He brought up his 50 with a check drive for two, and hit just his fifth boundary shortly after, but his stodgy innings was ended when he was trapped in front by Narine.
Taylor had reached his half-century by then too, and he joined with captain Brendon McCullum to close out the rest of the day.
The tourists, in a match they must win to square the three-Test series, lost their final four wickets in 26.2 overs before lunch.
Chanderpaul, who partnered Denesh Ramdin in a much-needed 200-run stand on the opening day, combined effectively with the tail to put on 60 for the final two wickets with Veerasammy Permaul (20) and Best (25).
Chanderpaul started the day six runs short of three figures but lost Sammy before reaching his milestone. Sammy edged a Tim Southee (four for 79) outswinger behind to wicketkeeper BJ Watling in the sixth over over of the day.
His departure prompted some caution from Chanderpaul before Southee provided just enough width for him to run past gully and to the ropes for his hundred.
Chanderpaul celebrated emphatically before kissing the Seddon Park pitch.
The left-hander had provided the backbone of the opening day's play with Ramdin, who reached 107, to claw the Windies away from trouble at 86 for five.
With Narine he was able to usher the score beyond 300 before the spinner was cleaned up by Trent Boult.
Permaul then hit an quickfire 20 and Best also swung the bat until Watling held on from Ish Sodhi minutes before lunch.
- Sports & Recreation
- Shivnarine Chanderpaul
- Darren Sammy
- Sunil Narine
- Hamish Rutherford