Otago were 173 for 5 at the close of play on the second day of the three-day match, needing another 127 to avoid the follow-on.
They lost two wickets in the last 20 minutes with the score on 159, with both Rutherford and Ian Billcliff getting out to poor shots after putting on 52 runs for the fourth wicket in just 40 minutes. The West Indies had reached 450 for 9 before their captain, Brian Lara, declared during lunch.
The visitors, weakened by the absence of the fast bowlers Courtney Walsh, Curtly Ambrose and Winston Benjamin, resorted to Rajindra Dhanraj's leg-spin in just the eighth over of Otago's innings, and he took 3 for 78 in 24 overs.
The openers Peter Dobbs and James Allan put on 62 for the first wicket but both went in the 30s, caught behind off Dhanraj and Benjamin respectively.
Dhanraj also bowled Martin Croy for 16 but both Rutherford and Billcliff took liberties with his sometimes erratic length.
With only one Test to his name, Dhanraj is hoping for a place in the first Test team along with Anderson Cummins, who made a century, and Kenneth Benjamin.
While Cummins was outstanding with the bat, his bowling was loose, limited by a minor back injury.
TOUR MATCH (Dunedin, NZ) Second day of three: West Indies 450 for 9 dec (J Murray 141no, A Cummins 107); Otago 173 for 5.Reuse content