The Henry problem has played a key role in understanding seawater intrusion into coastal aquifers and is a standard benchmark for coupled variable-density groundwater flow and solute transport. Physics-informed deep neural networks (DNNs) have been applied to a range of hydrogeological problems, but not yet to the Henry problem. The main challenge is that salinity affects the groundwater flow through density and viscosity, making joint training of flow and transport under a single loss prone to physically inconsistent solutions. We develop a physics-informed DNN framework for the classical Henry problem and the dispersive Henry problem. Inspired by Picard iteration, the framework uses two networks to represent the equivalent freshwater pressure head and salinity, and alternately trains one network while holding the other fixed. This turns each training step into a problem with fixed coefficients. A modified MLP with attention gates and residual-based adaptive refinement captures the steep concentration gradients in the transition zone. The more strongly coupled dispersive problem is solved through curriculum learning initialized from the converged classical solution. The DNN solution subjected only to the governing equations and boundary conditions matches the finite-element numerical solution very well for both problems. Compared with the finite-element solution with the same spatial discretization nodes, the salinity mean absolute errors are 2.56 × 10^-3 and 7.58 × 10^-3, with R² values of 0.9997 and 0.997 for the classical and dispersive Henry problems, respectively. The pressure-head R² exceeds 0.9999, and the saltwater toe agrees within 2%. Without alternating training, the transport residual becomes negligible, but the model collapses to the trivial c = 0 solution, increasing the salinity error by more than an order of magnitude. Alternating training is therefore necessary to obtain the correct solution. To our knowledge, this study presents the first physics-informed DNN solution of the Henry problem. It provides continuous, differentiable solution fields, and the alternating training strategy may serve as a reference for other strongly coupled flow-transport problems.