See Handbook 1 section 13.4.2 and 13.4.3. In short, the Ward Clerk must be have the Melchizedek Priesthood and be temple worthy. An assistant ward clerk must be temple worthy and must hold Melchizedek Priesthood if assigned to handle finances. I can't find anything definitive about whether branch clerks must hold the Melchizedek Priesthood but the
Branch Guidebook does say that if a branch president doesn't hold the Melchizedek Priesthood then the mission president handles finances so I would assume that any branch clerk that handles finances should have the Melchizedek Priesthood.