My first reaction is to problematize the notion of "relevance." There is no real reason that people should only experience things that are familiar and comfortable. The point of reading or watching plays from other eras is precisely to learn about people who have different belief and lives than we do. Thus in terms of staging, I think accurate historical reconstruction is a good way of letting the audience get a sense of experiencing something fundamentally different than the social media and local bubble of their daily lives. Thus I'd avoid a translation that used 21st century slang and instead look for a 19th century translation to emphasize that this is not a 21st century drama.
As part of this aims to make a production as much of a valuable experience of difference as possible, I'd avoid the tendency to make Nora a contemporary independent woman or an unambiguous heroine, and perhaps see if Torvald Helmer can be played sympathetically, so that the play becomes more about tensions in a marriage and the problems of deceit than simply the triumph of Nora in becoming an independent woman. Traditional proscenium staging, perhaps with a thrust, works well to give a sense of the respectable middle class setting which is such an important element of the play.
Creating a modern version of A Doll's House is quite possible and has been done in New York and London quite successfully. To consider that our civilization is beyond the issues that are presented in A Doll's House is to be blind to cultures throughout the world and the subcultures of what we consider modern society. Women and men all over the world exist in controlling marriages/relationships and it takes a great deal of courage and fortitude for them to leave. Psychological and physical abuse in relationships has not disappeared with the implementation of women's suffrage.
In producing the play modern devices may be used to replace the dated contrivances of the Victorian era. Use of cellphones, emails, and texts are easily inserted into the play without dramatically changing the scope of the characters, nor the theme of the play. With regard to direction, while Nora does come to an understanding of herself and the role she wants to take in society and her life, there is no need to be preachy. In fact the play's more dramatic scenes are best done quietly. In order to best relate with modern day women, it would be best to have the actors and actresses involved talk with psychiatrists regarding controlling and psychologically abusive relationships. The actress playing Nora would also benefit from talking with a woman who has made a similar decision to walk away from her marriage in order to "save herself."
The choice to leave the children with the husband seems as shocking today as it did in the Victorian period (especially if Torvald is portrayed as psychologically abusive and overly controlling). Nora, in the Victorian period, was viewed as selfish and a bad mother for leaving her children, according to the critics of the play. In the modern version, this choice would have to be handled carefully, at the risk of Nora being vilified for leaving her children with an abusive monster. However, it does bring up an interesting possibility for the play. If the play were to be gender swapped, the only way for the male version of Nora not to come off as a deadbeat would be to take the children with him when he leaves. This entertains a unique discussion that falls in line with Ibsen's initial points when the play was first produced. The question is if a man is in a psychologically abusive relationship, does he have the same rights to leave the woman? And what are his responsibilities regarding the children who, if left, will be in danger of the same abuse? Is he a deadbeat or is he simply standing up for himself? These questions were applied to Nora as the door closed at the end of the play and created a great deal of turmoil with critics, religious groups, and especially in the news and with the general public. It can be assumed a modern approach to the play would bring about similar results.