Countered spells don't resolve. A creature spell which is countered never enters the battlefield to become a permanent, so neither Enter nor Leave the battlefield abilities will happen, since it doesn't do either thing.
August 10, 2014 4:22 p.m.
The answer above is correct for your question, but I wanted to mention something else. Ashen Rider doesn't have a "leave the battlefield" effect, it has a "dies" effect. It won't trigger if the card is exiled or returned to your hand, for example. Just something to be aware of.
August 10, 2014 4:51 p.m.
Note also that "proc" is not a Magic term. Proc stands for Programmed Random Occurrence. Magic has triggers, which always occur at a prescribed time or event.