As long as you have less than around 750k DKK in the bank, you’re covered by an interbank insurance so don’t need to worry about losing your money.

If you’re just looking for a web bank with basic products, I would recommend Finansnetbanken.dk. It was recently bought by Jyske Bank but continues as a separate entity. In my 10+ years of staying with them, I’ve never been contacted by a rådgiver. As for the savings, you can just go to

http://www.mybanker.dk/sammenlign/opsparing/opsparingskonto/

and pick the one with the highest interest that satisfies your requirements. Note that some of them require you hold shares of the bank.

Personally, I keep my savings in an investment fund – for short term savings you can put them in a fund that invests in bonds that are about to expire (korte danske obligationer). That’ll usually give you a better yield than the banks with their fixed offers, and it’s much more flexible. Almost all Danish investments funds are ETFs so it’s pretty easy to put money into them from inside the web bank, and they’re organised through Investeringsforeningsrådet so have historical yields and risk statistics conveniently collected at ifr.dk.