Short answer: yes, usually, but not perfectly. A domed top does a simple job well: it sheds rain instead of letting water pool on a flat lid. That means less standing water on top, less overflow into the opening, and fewer drips running down the sides. In a busy city, where bins sit out in wind-driven rain, that shape can make a real difference.
But the dome is only one part of the design. If the opening is recessed or facing the wind, rain can still get in. If the liner bag tears, or if the bin has poor drainage, water can collect inside even when the lid stays dry. Wet waste and condensation can also create moisture that has nothing to do with the dome.
So I would say domed urban trash bins reduce rainwater buildup at the surface and cut down how much rain enters through the opening, but they do not guarantee a dry bin. The best designs combine a domed top with a sheltered opening, good drainage, and regular emptying and maintenance. Without those, the dome helps—but it is not a magic fix.