Buckskin Gulch is remote. The nearest town of any size is Kanab, Utah, about 40 miles to the north via US-89. Kanab has accommodation, fuel, and gear shops; this is where you should stock up before heading out. Page, Arizona is about 75 miles to the east and also has services.
From Kanab, drive south on US-89 toward the Utah-Arizona border. The Wire Pass Trailhead turnoff is near milepost 26 on the Arizona side. Follow House Rock Valley Road (unpaved) for about 8 miles to the trailhead. The road is passable in dry weather for two-wheel drive cars; high-clearance vehicles are recommended. Do not attempt it in wet weather — the road becomes hazardous and vehicles have become stuck for days.
There is no public transport to the trailhead. A rental car or private vehicle is essential. Las Vegas is the most practical major airport — about 3 hours away. Salt Lake City, Phoenix, and Denver are all within reasonable driving distance.