Quickly Googling both phrases brings up a number of results (mostly from sports websites) for each phrase; is one of the two in wider use and/or more correct?
I'm not sure you can/should compare them directly, because they're used differently. Running high can be used for any emotionally charged situation, good or bad. Riding high, on the other hand, usually has an implication of good emotions.
Edit: As requested in the comments, here are a few references from Google NGrams searching.
For Riding High:
Urbran Dictionary Definition - To be feeling very happy.
Florida is still riding high after the migration surge added 3 million residents and their investment dollars ...
He's riding high, thanks to a rebound in watch sales.
For Running High:
But with emotions running high on both sides, the abolitionists and the conservatives were on a collision course
With emotions running high, de Martel, fearing escalation into violence
With smoke blurring vision and emotions running high, inexperienced troops often fired on anyone wearing the enemy's color.
There are indeed a few Ngram instances of "emotions riding high" related to non-happy emotions, but they appear to be uncommon and I would say are mistaken usage.