Hi,
A Working Holiday permit to stay in Denmark is given for up to 1 year. Because it is a working holiday, you are allowed to work for up to 6 months, but you must not take a regular job. This means that your employment must be limited and not exceed the 6 months you are allowed to work during your working holiday.
With this restriction you will not be able to work for 1 year, which is a requirement for being eligible for Unemployment benefits from a Unemployment Insurance Fund (“A-kasse”) in Denmark.
Furthermore one of the conditions for your Working Holiday permit is probably that you must be able to support yourself during your stay,
https://www.nyidanmark.dk/en-GB/You-want-to-apply/Working-Holiday/Working-Holiday
This means that you can not receive public support, e.g. Unemployment benefits.
In other words – as a non-EU citizen – you will need a permanent residence permit before it is relevant to join an Unemployment Insurance Fund (“A-kasse”).
Let me know if you have other questions.
Best regards,
Anders Weber, A-kasser.dk