This is one of those that depends on which sanctioning body your game is being played under.
In all of them, a runner who is hit with a fair batted ball while in contact with a base is not out- unless he did something to intentionally interfere with the ball (like reach up and knock it down, when the ball would not have otherwise contacted him).
As a side note, this is totally different than in baseball, where the base offers no protection to a runner being hit by a batted ball.
That's the easy part. Now we'll throw a wrench into the ruling...
As already noted, in ASA softball the ball may become dead if it has not yet passed an infielder (other than the pitcher). In this case, the batter is awarded first base and any other runners would advance only if forced. If the ball has passed an infielder, then it remains live.
In USSSA, a fair batted ball that hits a runner in contact with the base always remains live.
In NSA, a fair batted ball that hits a runner in contact with the base is always automatically dead!
There you have three different sanctioning bodies and three completely different rules to cover the exact same play!