Natural gas hydrates have significant economic and environmental potential, and their exploitation and utilization are of strategic significance for national energy security and realizing “dual carbon goals”of peak carbon and carbon neutrality. The majority of hydrates in China are distributed in the South China Sea. The complexity and concealment of the marine environment result in highly uncertainty of engineering measurement data, which affects drilling safety. However, there is a lack of reliability analysis for wellbore stability in hydrate reservoirs. Based on the analytical model of wellbore stability in hydrate reservoirs and the specific geological conditions of the Shenhu area in the South China Sea, the reliability analysis methods including the advanced first-order second-moment method and the response surface method are employed to quantitatively assess the probability of wellbore instability during drilling, and to analyze the sensitivity of the wellbore instability probability and safe drilling fluid pressure window to the mean and uncertainty of the main parameters. The research results indicate that: (1) If the measurement data are accurate enough, the drilling in hydrate reservoirs in the Shenhu area of the South China Sea is highly safe and the safe drilling fluid pressure window is also large. However, the increased uncertainty of measurement data can significantly increase the probability of wellbore instability and narrow the safe drilling fluid pressure window. (2) A lower drilling fluid temperature can slightly reduce the probability of wellbore instability and significantly increase the safe drilling fluid pressure window. (3) The mean and uncertainty of the five main parameters have the same order of influence on the probability of wellbore instability, that is: initial in situ stress > initial internal friction angle > elastic modulus ratio > initial cohesion > initial elastic modulus. Accurate measurement of initial in situ stress in practical engineering can significantly improve the wellbore stability in hydrate reservoirs.