The typical person, right handed or not, would mount his noble steed from the left (non drive side) of the horse. Here's a good reason why from Answer.com:

"It is traditional to mount and dismount a horse from the left side the reasoning behind this goes back to when people carried swords, the swords mostly hung on the left hip they couldn't mount a right side because the sword got in the way so became traditional to mount from the left."

I'm sure you can mount a horse from the right, I've never met one but I found it interesting it does have history. And I always mount my bike from the left side, left over lessons from "Old Glue"

Scott