A Doll's House doesn't need a contemporary adaptation in order to be relevant to a modern audience; its themes already create the relevance. Torvald's unwillingness to take Nora seriously and his shame in her efforts to save his life resonate with issues today in which women's concerns are minimized.
Nevertheless, almost any good drama with universal themes can be adapted and given a more contemporary look. You could begin with costumes and setting. Other than the fact that they have household help, the Helmers are not much different than modern middle-class couples. Make their home a contemporary home with a television and a computer, and dress the Helmers in modern clothes and you've already hinted to the audience that their problems and issues could happen today. And the nurse could certainly be a modern-day "nanny" wearing casual clothes instead of the rigid uniform of the late nineteenth century.
Another way to adapt a drama to a contemporary setting is to use current music playing in the background and as the curtain rises. A clever director might look for songs that suggest some of the same themes as the play. Other stage elements such as lighting can enhance the contemporary setting, and the acting could play up the relationship among the characters in a more modern way. Even props around the house can indicate a contemporary lifestyle, such as current books, shopping bags from stores, and other accoutrements of modern life.
A New York Times review of a 2010 production (see link below) gives examples of just such contemporary touches; the article lauds the changes and describes their impact on the audience.