The monarchy is slightly controversial but the majority of danes like it. There are certain benefits - if we didn't have a king, we'd need a president instead who would be a much more politically divisive figure than the king is. As it is, the king is a much more uniting figure. We also don't need to have elections for the president or any of that stuff.
And no, of course he has no real power. Which is honestly good, cause a president would have more power than that. I personally prefer the situation as it is right now. The king stays because the people say so - that is also democracy.