The shortest route on-trail would be starting at the Freidlein Prairie Trailhead, using the Kachina Connector and part of the Kachina Trail (.3 miles and .4 miles respectively), and then 5.7 miles up the Weatherford Trail to the upper reaches of the Inner Basin Trail. From Snowbowl up the Humphrey's Summit Trail to the saddle, and then taking Weatherford Trail for 2.5 miles to where Weatherford meets the upper end of Inner Basin Trail is a slightly longer approach. The Humphrey's Summit Approach is 7 miles, so only about a half a mile longer, but the Weatherford is a more scenic and less heavily used trail anyway, so I definitely recommend going up Weatherford. Do consider the additional distance of whatever hiking you do once you reach Inner Basin Trail. It's two miles one way from the Weatherford-Inner Basin Trail junction to the popular "meadow" in the middle of Inner Basin. At a minimum I would estimate you are talking about a 17 mile with 3200 feet of elevation gain at least. If you are a backpacker, you might consider overnighting at Fremont Saddle, and then day hiking into Inner Basin, going over 1-2 nights.
I'm standing alone on the cliffs of the world
No one ever tends to me
Sitting alone, covered in rays
Some things are so my mind can breath