There are many banks in Denmark, large as well as small. Not all banks offer online banking in English, so we would advise you to enquire whether the bank of your choice offers such service, as it will make daily life much easier.
Remember to ask your advisor about the various fees for the services offered by the bank.
To find the bank with the lowest fees, you can use www.pengepriser.dk. The site is in Danish only. On the front page you can click on the family situation applicable to you - single, have a family or are a senior citizen (single or couple). Based upon your family situation and whether you rent (lejer) or own (ejer) your home, and your annual income before tax, the website will compare banks and suggest the cheapest bank that matches your selection.