Refused boarding to New Zealand because I couldn't demonstrate I'm an NZ citizen. What are my options?

You travelled with your daughter to Russia on Russian passports, but your daughter is a New Zealand citizen with a New Zealand birth certificate.

I assume that you have a valid New Zealand visa entitling you to re-entry into New Zealand.

You made the assumption that your daughter's New Zealand birth certificate would entitle her to re-entry into New Zealand, but the airline did not accept this when you attempted to check in for the flight from Russia to Korea (with an onward flight from Korea to New Zealand assumably) and refused you boarding.

Unfortunately, this requirement is specifically handled on the New Zealand Immigration website:

You will need an endorsement of your New Zealand citizenship in your foreign passport. An endorsement is a record that allows a New Zealand citizen to enter New Zealand as a citizen although they are travelling on a foreign passport.

That same page also states:

If you are overseas, your application must be sent before you travel to New Zealand.

From comments, you have indicated that you have had a conversation with NZ Immigration and they have told you to ask the airline to contact themselves but the airline is refusing - this is certainly something the airline is entitled to refuse to do, as their guaranteed information is provided to them via a system called Timatic, which is a database kept up to date providing immigration information for all countries.

In your case, Timatic has this to say for a New Zealand citizen flying on a Russian passport:

Russian Federation (RUS) - New Zealand (NZL) 29 Oct 2017

Visa or additional documentation is required for New Zealand.

I used the Skyteam interface.

At this point, you need to fix the documentation problem - the only way you can reasonably do that is to have New Zealand issue you with either an endorsement in your daughter's Russian passport (which you need a mailing address to achieve and this could take a week or more to achieve) or apply at a New Zealand embassy or commission for an Emergency Travel Document.

It would seem that applying for an ETD is the way to move forward in this situation, so:

If you’re overseas and don’t have a current passport and need to travel in a hurry you can apply for an emergency travel document.

https://www.passports.govt.nz/urgent-travel/

This will incur a $500 NZ fee and you need to apply at a New Zealand High Commission or embassy.

The relevant portion of New Zealand law which covers ETDs is worth quoting here, as it is actually fairly broad:

23 Issue of emergency travel document

(1) The Minister may, on application to the Minister in the form provided by the Secretary, issue an emergency travel document to any person where—

(a) there is reasonable cause to believe that the person is or may be a New Zealand citizen; and

(b) there is reasonable cause to believe that—

(i) the person’s passport has been lost or stolen or destroyed or is temporarily unavailable; or

(ia) the person does not hold a valid travel document; or

(ii) the person has applied for the issue of a passport and cannot supply sufficient evidence of New Zealand citizenship; or

(iii) an emergency has affected the availability of the information necessary to ascertain whether or not that person is already the holder of a New Zealand passport; and

(c) the person wishes to travel immediately, but, for reasons of passport security and integrity, the Minister considers that it is not desirable to issue that person with a passport.

Passports Act 1992

I have bolded the parts I feel are relevant - in your case, you have your daughter's New Zealand birth certificate, which would serve to provide reasonable cause to believe she is a New Zealand citizen.

Alternatively, contact New Zealand government overseas traveller helpline and discuss your issue with them:

If you're overseas and need help now, call +64 4 439 8000.

With regard to your financial hardship situation, this can be particularly distressing - unfortunately there is little that can be done in this situation.

The New Zealand government will not lend you money - they will however help you to contact friends and relatives in New Zealand that may be able to help:

As a New Zealand citizen overseas, you have no right or claim to financial assistance from the New Zealand Government.

...

We can help by:

• Assisting you to contact family or friends to help you in transferring funds, if you are having trouble making contact with them.
• Transferring funds through the Ministry or via one of our embassies when there is no other option, for a fee.
• Discussing other ways of resolving financial issues if none of the above options are successful.

We cannot:

• Pay your hotel, travel or other bills, legal or medical expenses or costs of returning a body to New Zealand.

• Give you money.

https://www.safetravel.govt.nz/financial-difficulties

As you were denied boarding for insufficient travel documentation, the airline may not be under any obligation to refund your ticket or allow you to use its value against a rebooking. They may also charge you a change fee if they do allow you to use the value. This is dependent on the airlines Conditions of Carriage, and also the local laws of New Zealand and Russia.

You may be in a better situation than other travellers however, as you and your daughter are citizens of the country that you are stuck in - you may be able to seek emergency assistance from that country, such as emergency accommodation etc.

This won't help you to get back to New Zealand, but it may stop you from starving or sleeping on the streets.

Regarding responsibilities in this situation, it is the travellers responsibility for ensuring they have an entitlement to re-enter the country of origin for the trip, and it is also the travellers responsibility for ensuring they have sufficient proof of that entitlement.

JonathanReez has mentioned a third option in the comments, in that your daughter may be able to apply for a New Zealand entry visa on her Russian passport - and while this is indeed a potentially viable option, it may not be straight forward or easy to accomplish.

Your daughter would have to meet the criteria for a visitors visa on her own merits, and any application may take anything from several weeks to months depending on the visa applied for (current average processing time for a Visitor visa is 20 days for an online application).

You may encounter issues as an NZ citizen applying for an NZ visa (even on a foreign passport), as well as any issues surrounding whether she would be eligible to stay in NZ past the visa expiration date without incurring issues on her Russian passport further down the line - that is something you seriously need to discuss with NZ immigration if you go this route. The easiest option there, once you are back in NZ through using this option, is for your daughter to apply for an NZ passport, leave the country on the Russian passport and re-enter the country on the NZ passport.

As is evident with the recent case of the Australian deputy prime minister, it is possible to renounce your New Zealand citizenship. Therefore, even though a New Zealand birth certificate may provide a strong suggestion that your daughter still holds this citizenship it is not proof. For example, she could have renounced it if she is of legal age. Furthermore, the mere fact of having a New Zealand birth certificate does not mean you acquired New Zealand citizenship at birth.

The Timatic interface linked in Moo’s answer allows you to enter the documents (Russian passport, New Zealand birth certificate) and travel route you have and it clearly states that:

Birth Certificate issued by New Zealand to a national of New Zealand is not listed as an accepted document by New Zealand. Please check the document details have been entered correctly.

(It will only give this message if you do not enter the Russian passport.)

The only documents that reasonably prove citizenship beyond doubt are a passport and ID cards; the latter only if they exist at all (not all countries issue them) and if they mention citizenship. Without either of these documents the airline must assume that you have insufficient documentation and thus should refuse to board you. If they decide to let you board nonetheless and you are deemed inadmissable at New Zealand immigration due to insufficient documentation that they did not check they will be fined a lot and potentially face other legal consequences. No airline in their right mind would do that.

Everything else is sufficiently covered in the other answer by Moo.