Low Earth Orbit (LEO) satellite networks play a crucial role in achieving the ubiquitous connectivity. Due to the limited radio resources and large moving coverage area, the access procedure of LEO networks exhibit a dynamic nature. Traditional access class barring (ACB) scheme adopted in terrestrial networks is able to deal with the access conflicts of massive terminals, but is not suitable for LEO networks with fluctuated serving terminals. To this end, a dynamic access control based on reinforcement learning (DACRL) is proposed in this paper to improve the access adaptation of LEO networks. DACRL iteratively conducts the access terminal quantity adjustment and access conflict control. To achieve a fine access terminal quantity adjustment, a deep Q-network (DQN) with state windows is designed to generate the proper access parameters without knowing the number of access users. To relieve the conflict access, reward of the proposed DQN is formulated by obtaining the tradeoff between the access success rate and the average access delay. Simulation results show that, compared with the reference algorithms, DACRL significantly improves access efficiency and is able to satisfy the access requirements of dynamic terminal quantity.