The radiation in space isn't like the radiation here on earth. It is VERY low dose, so the only way you can get sick from it is if you stay in it for a month or more. Obviously the Apollo flights never took that long.

Also, if a solar flare did pop up, they have hours, even days before detection of the flare and the time it reaches earth. By that time the astronauts are either out of harm's way, or they turn the ship so that the bulk of the landing module faces the flares, then the bulk of the ship plus the fuel will hold off the radiation.