Soft robotics is an emerging technology with excellent application prospects. However, due to the inherent compliance of the materials used to build soft robots, it is extremely complicated to control soft robots accurately. In this paper, we introduce a data-based control framework for solving the soft robot underwater locomotion problem using deep reinforcement learning (DRL). We first built a soft robot that can swim based on the dielectric elastomer actuator (DEA). We then modeled it in a simulation for the purpose of training the neural network and tested the performance of the control framework through real experiments on the robot. The framework includes the following: a simulation method for the soft robot that can be used to collect data for training the neural network, the neural network controller of the swimming robot trained in the simulation environment, and the computer vision method to collect the observation space from the real robot using a camera. We confirmed the effectiveness of the learning method for the soft swimming robot in the simulation environment by allowing the robot to learn how to move from a random initial state to a specific direction. After obtaining the trained neural network through the simulation, we deployed it on the real robot and tested the performance of the control framework. The soft robot successfully achieved the goal of moving in a straight line in disturbed water. The experimental results suggest the potential of using deep reinforcement learning to improve the locomotion ability of mobile soft robots.