With the development of IoT technology and 5G massive machine-type communication, the 3GPP standardization body considered as viable the integration of Narrowband Internet of Things (NB-IoT) in low Earth orbit (LEO) satellite-based architectures. However, the presence of the LEO satellite channel comes up with new challenges for the NB-IoT random access procedures and coverage enhancement mechanism. In this paper, an Adaptive Coverage Enhancement (ACE) method is proposed to meet the requirement of random access parameter configurations for diverse applications. Based on stochastic geometry theory, an expression of random access channel (RACH) success probability is derived for LEO satellite-based NB-IoT networks. On the basis of a power consumption model of the NB-IoT terminal, a multi-objective optimization problem is formulated to trade-off RACH success probability and power consumption. To solve this multi-objective optimization problem, we employ the Non-dominated Sorting Genetic Algorithms-II (NSGA-II) method to obtain the Pareto-front solution set. According to different application requirements, we also design a random access parameter configuration method to minimize the power consumption under the constraints of RACH success probability requirements. Simulation results show that the maximum number of repetitions and back-off window size have a great influence on the system performance and their value ranges should be set within [4, 18] and [0, 2048]. The power consumption of coverage enhancement with ACE is about 58% lower than that of the 3GPP proposed model. All this research together provides good reference for the scale deployment of NB-IoT in LEO satellite networks.