Mining of the extensive, steeply dipping ore deposit takes several decades. An open-pit mining method is more often used in the early years of such a mining enterprise (ME). The management of the enterprise is faced with the problem of changing the mining method as the depth of the quarry increases. Untimely solution of this issue or the choice of the wrong strategy for the development of ME leads to a decrease in profitability, and the emergence of environmental and social difficulties. We studied the functioning of one hundred and seven MEs from different countries and substantiated four main sustainable development strategies for ME and its main system—the open-pit mining and technical system (MTS): adjustment of the current stage of mining indicators, transition to a new stage of mining, transition to a combined open–underground mining, and mine closure. The result of our research is an original methodology for selecting a strategy for MTS sustainable development. Our methodology is based on a new system of parameters and indicators for evaluating the sustainability of the opening-up of an opencast system (OOS). This assessment system includes twenty-three indicators that characterize the technical, technological, economic, social, and environmental factors of sustainable development. We propose to select a strategy for MTS sustainable development using combined fuzzy AHP-MARCOS multicriteria decision method (MCDM). The result of our case study for the Malyi Kuibas ore deposit was the choice of a mine closure strategy. The reliability of the obtained result is confirmed by a multilateral sensitivity assessment using nine other known MCDMs, while changing the criteria weights and composition of strategies. The results of the study prove the need for a timely decision to change the MTS development strategy as the depth of production increases. In addition, we have shown the effectiveness of the selection methodology based on the multicriteria assessment of the OOS sustainability.