Cloud computing has become one of the most important technologies for reducing cost and increasing productivity by efficiently using IT resources in various companies. The cloud computing system has mainly been built for private enterprise, but public institutions, such as governments and national institutes, also plans to introduce the system in Korea. Various researches have pointed to security problems as a critical factor to impede the vitalization of cloud computing services, but they only focus on the security threats and their correspondents for addressing the problems. There are no studies that analyze major security issues with regard to introducing the cloud computing system. Accordingly, it is necessary to research the security factors in the cloud computing given to public institutions when adopting cloud computing. This research focuses on the priority of security solutions for the stepwise adoption of cloud computing services in enterprise environments. The cloud computing security area is classified into managerial, physical and technical area in the research, and then derives the detailed factors in each security area. The research derives the influence of security priorities in each area on the importance of security issues according to the identification of workers in private enterprise and public institutions. Ordered probit models are used to analyze the influences and marginal effects of awareness for security importance in each area on the scale of security priority. The results show workers in public institutions regard the technical security as the highest importance, while physical and managerial security are considered as the critical security factors in private enterprise. In addition, the results show workers in public institutions and private enterprise have remarkable differences of awareness for cloud computing security. This research compared the difference in recognition for the security priority in three areas between workers in private enterprise, which use cloud computing services, and workers in public institutions that have never used the services. It contributes to the establishment of strategies, with respect to security, by providing guidelines to enterprise or institutions that want to introduce cloud computing systems.