Kembali
Basham is part of a back-row reshuffle with James Botham starting instead of fellow Cardiff flanker Alex Mann, and Jac Morgan completing the trio.
Wing Josh Adams, who scored two tries in the 28-21 victory against Japan, is left out in favour of Leicester wing Gabriel Hamer-Webb who wins only his second cap.
Lock Adam Beard comes in for Ben Carter as head coach Steve Tandy makes four starting changes.
Sale prop Tomas Francis has been ruled out for the rest of the campaign by injury with Cardiff tight-head Keiron Assiratti called into the squad.
Back rower Taine Plumtree and prop Gareth Thomas are named among the replacements with Rhys Carre not in the match-day 23.
Leicester back-rower Wainwright has proved Wales’ most effective forward ball-carrier in recent times.
He picked up a minor calf injury against Japan and has spent the week in a protective boot as he battles to be fit against Australia
Basham was recalled to the Wales squad after impressing for Cardiff.
His previous Wales appearance came as a replacement in the 2024 Six Nations against England, while his past international start came in the 2022 tournament against the same opposition.
It is only his second Test start at number eight.
"I'm delighted that Taine gets an opportunity this weekend," said Tandy.
"It's been a while since his last appearance for Wales but he didn't give up and has put in a lot of dedication and hard work to get back.
"The way he's been playing for Cardiff has been great and I'm excited to see him go on Saturday."
It will be a second cap for Hamer-Webb after he started the Six Nations defeat against Scotland in February.
The 26-year-old wing has been in fine form for his club side having scored a hat-trick against Saracens in the Prem.
Adams, 31, had made only one start for Cardiff before facing Japan where he had a major effect on the game with two tries and a turnover.
Beard, 30, will win his 69th cap after playing for his French club Montpellier last weekend.
Wales are looking to improve the line-out after losing six on their own throw against Japan.
Wales have not beaten New Zealand since 1953 with the All Blacks winning the past 34 matches and scoring more than 50 points on their three previous trips to Cardiff.
Dave Rennie’s side resumed their Nations Championship campaign with an impressive 43-21 victory against Scotland in Edinburgh last weekend.
"We're excited for the opportunity we have against New Zealand," said Tandy.
"It was good to get the win last Saturday, but some elements of our execution weren't where we wanted.
"We have to go up another level this weekend and be more ruthless. We have one of the best teams in the world coming.
"We want to be better and put in an 80-minute performance."
Replacements: Elias, G Thomas, D Lewis, Carter, Plumtree, Morgan-Williams, Edwards, Hennessey.