Yes, common fishing rods being sold as of 2023 have no problem getting wet from the rain or getting dunked in water. Fishing rods have been made from a wide variety of materials throughout history, so a particular rod might be damaged or need care after major contact with water. But for modern fishing rods constructed of materials like graphite and fiberglass water is a non-issue. Saltwater may cause metal components like guides to corrode but generally speaking, modern rod components have very good anti-corrosion properties.
For cleaning after contact with freshwater, a simple wipe down with a rag or left alone to air dry is sufficient. For post-saltwater use, clean by using freshwater (removes salt residue) and then dry. In either case, rod cleaner can be applied after the rod is dry to reduce the chance of corrosion and provide some extra cleaning power.
Fishing rods should be stored indoors if possible but if left outside and exposed to rain and elements for short periods should still be okay. A fishing rod permanently stored outside will likely see damage over time from UV from the sun, temperature changes, and flex on the rod blank.
Fishing Fundamentals