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In recent years, magnetoactive soft continuum robots (MSCRs) with multimodal locomotion capabilities have emerged for various biomedical applications. Developments in nonlinear dynamic models and effective control methods for MSCRs are deemed vital not only to gain a better understanding of their coupled magneto-mechanical behavior but also to accurately steer the MSCRs inside the human body. This study presents a novel dynamic model and model-based AI-driven control method to guide an MSCR in a fluidic environment. The MSCR is fully exposed to fluid flows at different rates to simulate the biofluidic environment within the body. A novel nonlinear dynamic model considering the effect of damping and drag force attributed to fluidic flows is first developed to accurately and efficiently predict the response of the MSCR under varying magnetic and mechanical loading. Fairly accurate correlations were observed between the theoretical responses based on the developed magneto-viscoelastic model and the experimental data for various scenarios. A novel model-based control algorithm based on a fractional-order sliding surface and deep reinforcement learning algorithm (DRL-FOSMC) is subsequently developed to accurately steer the magnetoactive soft robot on predefined trajectories considering varying fluid flow rates. A fractional-order sliding surface and a compensator, trained using the deep deterministic policy gradient algorithm, are designed to mitigate the amount of chattering and enhance the tracking performance of the closed-loop system. The stability proof of the developed control algorithm is also presented. A hardware-in-the-loop experimental framework has been designed to assess the effectiveness of the proposed control algorithm through various case studies. The performance of the proposed DRL-FOSMC algorithm is rigorously assessed and found to be superior when compared with other control methods.