Getting there is always an issue. We fly to Fairbanks and rent a car, then leave it at the trailhead. However, leaving a car is always an gamble because cars occasionally won't start at 40 below, and then you're stranded. The trailhead is an hour north of town and taxis are probably very expensive, but perhaps cheaper then renting a car for a whole week. However, you will want to pre-arrange your ride home. You might not be able to call a taxi upon return, as cell phone reception is spotty to nonexistent out there.
Best to fly into Fairbanks. Anchorage is a seven-hour drive from Fairbanks, so eight hours to the White Mountains. Alaska is not a small state.