It depends on whether the collecting activity is commercial or noncommercial in nature and subject to river-administering agency regulation. Mining under the 1872 mining law is a commercial and business activity tied to valid existing rights of claims and is regulated as such (36 CFR 228, 43 CFR 3809, 8365, et al).
Non-commercial mineral collecting for recreational purposes (e.g., hobby collecting, rock-hounding, gold panning, sluicing, or dredging) may be authorized by the Bureau of Land Management or the U.S. Forest Service depending on the amounts collected, size and scale of activity, resource values impacted, and river management objectives. This collecting is subject to state, local, and other federal regulations.
The National Park Service and the U.S. Fish & Wildlife Service generally prohibit both commercial and non-commercial locatable mineral collecting (subject to valid existing rights). For refuges in Alaska, the FWS under 50 CFR 36.31(b) allows surface collection by hand of gold (including handheld gold pans) for recreational use only; however, collection involving surface disturbance (e.g., the use of shovels, pick axes, sluicing or dredging) is prohibited.