In terms of common consumer exports, the most common ones are pork products - a lot of which are baby back ribs and ham and bacon, come from Denmark. Other products are various cheeses and beers (Carlsberg and Tuborg being the main ones). Lego toys is a Danish company. B&O is an upscale audio manufacturer. I think the coffee company Gevalia is Danish.
Oh - and of course those damn butter cookies! If you like them, look for the "real" brand, which I think is called "Kjeldsen" ... a lot of the other "Danish butter cookie" brands are okay, they are not as good as Kjeldsen.
If you're looking for fine china, then the Royal Copenhagen brand really is one of the world's absolutel finest china manufacturers, and of the few that still makes hand-painted china. Their Flora Danica line is so expensive it is absolutely ridiculous
- but their Blue Flower, or Fluted, or Mega patterns are very nice china, and considering they are individually hand-painted in Denmark (with the most expensive labor costs in the world) they are reasonably inexpensive.
In addition that there are lots of industries that the average consumer won't necesarily run into - like shipping (Maersk) and wind power generation, and all kinds of niche electronics markets.
I'm sure I am forgetting some.
ETA: Georg Jensen is a famous Danish silversmith, so if you are insanely wealthy, you can buy some George Jensen silverware ("acron" is the classic, and nicest pattern, IMO - but their modern stuff is nice too).
As a warning - Keep in mind that the Carlsberg beer you most often find in the U.S. may be "imported" but it is not made in Denmark (but under license in a Canadian brewery), and doesn't quite taste like Danish Carlsberg. But I suppose the Carlsberg company gets some of the profits. If you can find it, Tuborg is better beer than Carlsberg, and is really good beer!