In terms of concentration, I would say no. With higher rainfall and snowpack melt, you will have more solute and therefore a greater ability to dissolve saline species. However, you also have more water so the concentration of salts in the rivers shouldn't show a dramatic change (unless the river begins to run through a salt flat or something like that).
To make the rivers more saline, you would need to change other factors such as temperature, pressure, and input/output from the system. Since we're working with a river, pressure remains relatively constant at atmospheric conditions. For the relevant temperatures (0 C - 30 C), the solubility of NaCl in water changes from 35.65 g NaCl/100 mL water to 36.09 g NaCl/100 mL water. That's a ~1.2% change, which means temperature does not have a large effect over that range. This leaves us with the input and output from the system. Without an outlet, the water level will fluctuate between the effects of input and evaporation. Meanwhile, the salinity will only increase as more salt species are dissolved and introduced. Therefore, as long as the river keeps moving, its salinity will not change significantly.