I was told I have a criminal record that could block my visa

It can, but it depends on the sentence and the nature of the offence. The Immigration Act 2009 and the character requirements in Immigration New Zealand's instructions generally require applicants to be of good character. Certain convictions, such as sentences of imprisonment of a specified length, can make you ineligible or lead to a prohibition on entry for a period, but there may be discretion to grant a waiver in some cases. Drink-driving and other driving convictions can also count, so be honest about your history in your application, because failing to disclose can lead to a declined application or a ban for false information. Obtain certified records, and a licensed adviser can assess whether you are likely to be eligible or need to request a character waiver.