The atmospheric water vapor is commonly monitored from ground Global Navigation Satellite System (GNSS) measurements, by retrieving the tropospheric delay under the Zenith Wet Delay (ZWD) component, linked to the water vapor content in the atmosphere. In recent years, the GNSS ZWD retrieval has been performed on shipborne antennas to gather more atmospheric data above the oceans for climatology and meteorology study purposes. However, when analyzing GNSS data acquired by a moving antenna, it is more complex to decorrelate the height of the antenna and the ZWD during the Precise Point Positioning (PPP) processing. Therefore, the observation modeling and processing parametrization must be tuned. This study addresses the impact of modeling on the estimation of height and ZWD from the simulation of shipborne GNSS measurements. The GNSS simulation is based on an authors-designed simulator presented in this article. We tested different processing models (elevation cut-off angle, elevation weighting function, and random walk of ZWD) and simulation configurations (the constellations used, the sampling of measurements, the location of the antenna, etc.). According to our results, we recommend processing shipborne GNSS measurements with 3° of cut-off angle, elevation weighting function square root of sine, and an average of 5 mm·h−1/2 of random walk on ZWD, the latter being specifically adapted to mid-latitudes but which could be extended to other areas. This processing modeling will be applied in further studies to monitor the distribution of water vapor above the oceans from systematic analysis of shipborne GNSS measurements.