NZ Post staff are working with affected customers who have made contact.
NZ Post has also notified the Privacy Commissioner's office.
"Following a full investigation by NZ Post, disciplinary action was taken, which resulted in the worker being stood down.
"The matter may still be referred to the police," Mario said.
"We are extremely disappointed that on this occasion, the trust we place in our employees was not met, nor importantly, our customers' expectations.
"We ask any customers who think they may have been impacted by this to contact their senders and NZ Post."
If you have a specific question arising from this incident or would like to make a further complaint, contact NZ Post's Customer Care Centre via its website www.nzpost.co.nz, or phone 0800 501 501.