Residents of most European countries and some countries from other parts of the world can now enter Denmark, provided they have booked a minimum of 6 nights in Denmark (does not apply for Norwegians). See here if you are eligible to enter Denmark at the moment.
Updated July 3, 2020.
Gradually, Denmark's borders are opened for travellers from countries that meet certain criteria, including a certain number of covid-19 patients per capita. At the moment, tourists from most European countries as well as Australia, Canada, Japan, New Zealand, South Korea and Thailand can enter Denmark. Unless they show clear signs of illness.
Stay updated on the current status
Worthy purpose and rules regarding entry
Travellers from countries defined as banned on the list above must expect to be denied entry at the Danish borders, including airports, unless they have a worthy purpose for entering. See what is considered a worthy purpose by the Danish authorities in this Q&A which is updated continuously.
Minimum 6 nights stay
To enjoy holiday in Denmark, visitors must prove they have booked a minimum of 6 nights in Denmark (not necessarily the same place). This rule does not apply if you reside in Schlesvig-Holstein, Skåne, Halland or Blekinge.
Joint Danish Authorities Hotline: +45 7020 0233.
As of June 15 2020 new guidelines for safe travel became effective in Denmark's airports. Besides including compulsory medical face masks at the terminals, the guidelines issued by The European Union Aviation Safety Agency (EASA) apply to all European airports and include a set of things you need to consider before flying. See this practical guide by Copenhagen Airport.
Danish Health Authorities' information for tourists: