In the face of the discourse about the democratic deficit and declining public support for the European Union (EU), institutionalist scholars have examined the roles of institutions in EU decision making and in particular the implications of the empowered European Parliament. Almost in isolation from this literature, prior research on public attitudes toward the EU has largely adopted utilitarian, identity and informational accounts that focus on individual‐level attributes. By combining the insights from the institutional and behavioural literature, this article reports on a novel cross‐national conjoint experiment designed to investigate multidimensionality of public attitudes by taking into account the specific roles of institutions and distinct stages in EU decision making. Analysing data from a large‐scale experimental survey in 13 EU member states, the findings demonstrate how and to what extent the institutional design of EU decision making shapes public support. In particular, the study finds a general pattern of public consensus about preferred institutional reform regarding powers of proposal, adoption and voting among European citizens in different countries, but notable dissent about sanctioning powers. The results show that utilitarian and partisan considerations matter primarily for the sanctioning dimension in which many respondents in Austria, the Czech Republic, Denmark and Sweden prefer national courts to the Court of Justice of the EU.