Hours after the first Test ended at Bay Oval, the 24-year-old had tweeted: "A bit disturbing hearing racial insults today whilst battling to help save my team. The crowd was been amazing this week except for that one guy."
Archer also said that the fan had later contacted him on Instagram.
New Zealand Cricket says its investigation has failed to find the spectator responsible - despite reviewing CCTV footage, audio and social media posts, and interviewing bystanders.
It has now passed over all of the information it has gathered to the police.
NZC chief executive David White said: "What happened to Jofra was reprehensible and has led to a general upscaling of security around the area of racial abuse at all our international venues.
"Should the person responsible ever reoffend, we believe we have enough information to link him to the Bay Oval incident."
Mr White has also vowed that the spectator responsible for the abuse would be banned "from all its international venues for a lengthy period" if he was conclusively identified.
Archer, who burst on to the international scene during the summer as England won the Cricket World Cup, had disappointed with the ball - taking just 1-107 from 42 overs.
The tourists were bowled out for 197 as they fell to an innings and 65-run defeat at Mount Maunganui, losing a flurry of wickets to collapse from 121-4 to 138-8.
New Zealand won the series 1-0 as the second Test in Hamilton was drawn.