How the cost of living compares depends a lot on where you're coming from, and what your lifestyle is like. I would say overall it's cheaper than the SF Bay Area, though the first few prices a tourist sees are likely to be shockingly expensive ($8 coffee, $10 beer). The main thing that is more expensive is eating out, mostly because there is less income inequality (even McDonald's employees make ~$40k/yr). Food at supermarkets is roughly the same price, maybe a bit more expensive. Rent is much, much cheaper; I have a really nice 2-bd apartment in a great walkable area, near the city center and with a view, and it's under $1500/mo. Transit depends on your lifestyle; it would be much more expensive to own a car, but I found I didn't need one anymore, because public transportation is very good, and you can walk/bike most anywhere anyway. So my transit costs went down, but those of others might well go up.
Good quality of life I'd say. Lots of parks, nice city, interesting neighborhoods, good work/life balance, low crime, fairly multicultural. Easy to get by in English as well (even when dealing with the government or healthcare). The large role of English in public life surprised me a bit; it's not just that people speak English, but many public events are also held largely in English (especially at museums, universities, festivals, hackerspaces, etc.). That makes it a fairly easy place to meet non-Danish Europeans, because there are a lot of them and they come to such events. So I've met a bunch of Germans, Spaniards, Italians, Poles, etc. who I doubt I would've met in their own countries. On the other hand, the general English-friendliness of Copenhagen makes it harder to learn Danish, or to meet Danes who aren't also part of the mixed-nationality social scene (especially those who aren't part of the academic or tech scenes, which are more internationally oriented than society in general is).
Biggest downside imo is that it's gray and overcast about 6-7 months a year, and for 3-4 of those months the days are also very short. On the plus side, people as a result really appreciate nice days, and almost literally the entire city will be outdoors grilling and lounging in parks on nice summer days, with a palpably positive mood.