The paper highlights the connection between the European Union banking system and a set of representative factors—macroeconomic, social, and governance factors—selected from the perspective of sustainability and well-being. The analysis is carried out as a panel regression on EU member countries with annual data for 2005–2018, and it explores the impact of the selected factors on the robustness of the banking systems in the European Union countries. The analyzed variables to reflect the robustness of the banking system were the domestic credit to the private sector and the nonperforming loans (NPL) rate. Those indicators are of high relevance and concern within the current pandemic context. The results show that the banking development degree influences the increase of private-sector lending and the decrease of the NPL rate. Social and governance factors impact differently the level of private sector and NPL rate. All macroeconomic indicators used to influence the level of private-sector lending. The research reflects the fact that to promote and adopt a culture of sustainability and to ensure well-being, a close collaboration between all sectors of an economy is needed, together with a strong policy interconnection and harmonization between micro and macro.