Let's read the directive, Article 4:
(7) ‘controller’ means the natural or legal person, public authority, agency or other body which, alone or jointly with others, determines the purposes and means of the processing of personal data; where the purposes and means of such processing are determined by Union or Member State law, the controller or the specific criteria for its nomination may be provided for by Union or Member State law;
‘processor’ means a natural or legal person, public authority, agency or other body which processes personal data on behalf of the controller;
So, the authors of distributed software might constitute "controllers", and every single node in the network is a "processor".
Some people will jump to the conclusion that this renders blockchain technology completely incompatible with GDPR, but I think at this point an argument can be made about what constitutes a legitimate basis. That gets out of questions of withdrawal of consent.
See "whereas" clause 47:
The legitimate interests of a controller, including those of a controller to which the personal data may be disclosed, or of a third party, may provide a legal basis for processing, provided that the interests or the fundamental rights and freedoms of the data subject are not overriding, taking into consideration the reasonable expectations of data subjects based on their relationship with the controller. Such legitimate interest could exist for example where there is a relevant and appropriate relationship between the data subject and the controller in situations such as where the data subject is a client or in the service of the controller. At any rate the existence of a legitimate interest would need careful assessment including whether a data subject can reasonably expect at the time and in the context of the collection of the personal data that processing for that purpose may take place.
In which case you'd need a sufficient warning to the user that all actions on a blockchain are irrevocable, plus a declaration that the user is effectively a co-controller of the data.
You could also contact the EU Blockchain Observatory for an opinion?