I need three 20a circuits in basement. I planned to run #8 wire 50feet from main panel to a new sub panel in basement where I would have 3 20a breakers. Will this be OK or do I need #6 wire for the feed to the subpanel?

The #8 wire will be good for 40 amps, assuming Romex cable. I think this will be plenty for the 3 new circuits plus a bit more. If you run #6 Romex, the sub panel can be 50 amps.