Fastpitch Softball: Run-Basing, RBIs, and Earned Runs Explained

Question
A baserunner is on base because of an "error." If the next batter gets a base hit, and the runner scores, is it a RBI for that batter.

Answer
Yes it is an RBI.  It really doesn't matter how the runner got on base, it matters how they are advanced.  

For the pitcher is this case, if that runner scores, it isn't an earned run.  This is because the defense blew the opportunity to put the runner out, which isn't the pitchers fault.