I think this is the most important misconception GB always had about the EU. It's a union which means, having a well tempered partnership should be the goal - not cherry-picking like the UK often did.
Or, to resemble a famous quote: Ask not, what the EU can do for you - ask what you can do for the EU!
We are not ready for further expansion of the EU project.
If this causes the EU to split in two parts: A smaller but stronger EU politcal/economical union and a broader but weak trade cooperation amogst european countries. Then Denmark will probably end up outside the political union.
We too had a number of referendums, more than one voting "no to EU" but always our representive polticians chose to negotiate a deal and create a new referendum which just scraped a yes.
The UK being out adds legitimacy to the "being out" case which EU opponents in Denmark and other places will use.
The dissatisfaction with the EU will much more visibile after this.
Getting out of the EU will not solve any of that. It could be that isolated countries will do better economicaly, and will be able to isolate themselves from the problems happening around them. It could be that you stay rich, or become richer, by closing your eyes and ears to the suffering happening around you, while the rest are left carrying the burden.
How sad.
The EU is the chance of Europe to be heard around the world, a great project being squandered by some right wingers opportunists.
That said, I'm certainly not that happy with the EU myself (also Dane), and really think it needs to shape up, but that is more related to the abundance of literally retarded regulations that they force down upon member countries.