Personally, though I am hugely in favour of the EU, I think the democratic deficit, both perceived and actual, is a real problem and should be addressed by the EU. While there are elections to the European Parliament, turnout is so low across most countries that I think it's a stretch to call it representative. The vast majority of citizens do not know who their representatives are, what issues they are voting on, or how to contact them. That's not democracy and it needs to be fixed.
In a previous job, I had to work with one of the EU Commissioners on developing a new policy area. I came away from that experience with the view that the way EU legislation is developed is FAR from a democratic process.
I also think that too often the public has been given the explanation that rules cannot be changed 'because it's EU law'. One example we saw in Britain was the so-called 'tampon tax', where a grassroots campaign from female rights organisations to abolish VAT on women's sanitary products was denied despite widespread support. The government said it wasn't allowed to fix the problem because 'it's EU law', which is true. I think that leaves people feeling a democratic unfairness because they don't have a clear path to influence legislation, campaign or vote for change on such issues.
While none of this is the main reason a majority voted to leave the EU (immigration was by far the most important issue), I think the EU needs to seriously consider improving how its citizens can feel heard and represented, and allow more flexibility and localism.