Makes me sneeze more than usual. Never had a nose bleed though. I think some sort of vertical climb in a fighter might do that, not a B777!
Cosmic radiation has been measured for years and never done anyone much harm. Given the lifestyle of most jet pilots I reckon the subsequent thrashing in the bar is more harmful to the body than any high altitude cosmic stuff. Droopy nose drivers were tested for years and dispite heavy exposure to the stuff they suffered no ill effects (not sure how it effected their offspring though!).
Generally I find the sneezing stops if I drink plenty of water in the cruise. I've found the quantitiy of water I need to keep sufficiently hydrated on the 777 is greater than previous types.