Without knowing where in the West you want to hike, this could be a very long list. Here are some ideas near places I feel have good day hiking based on personal experience.
I haven't actually stayed at all these places, and some have multiple lodge/camp-type accommodations. For example, Yosemite Valley has everything from the five-star Ahwahnee to the rustic Housekeeping camp, and South Lake Tahoe has numerous accommodations, from cheap motels to private cabins/houses for rent.
o Yosemite Valley (Yosemite NP/Yosemite Valley)
o Tuolumne Meadows/White Wolf (Yosemite NP/high country)
o Tioga Pass Resort (Yosemite NP/high country)
o Evergreen Lodge (Yosemite NP/Hetch Hetchy area; Emigrant Wilderness)
o Wawona (multiple; Yosemite NP/Mariposa Grove)
o Lee Vining (multiple, Yosemite NP/Hoover/Ansel Adams Wilderness)
o June Lake (multiple, Yosemite/Ansel Adams/Devils Postpile)
o Twin Lakes/Mono Village (northern Yosemite/Hoover Wilderness)
o Wuksachi Lodge (Sequoia NP)
o Monticito Lodge (Giant Sequoia NM)
o Silver City (Sequoia NP/Mineral King)
o Cedar Grove (Kings Canyon NP/Kings Canyon area)
o Grant Grove/Wilsonia (Kings Canyon NP/Grant Grove area)
o Mammoth Lakes (Yosemite/Devils Postpile/Ansel Adams Wilderness)
o Rock Creek Lodge (John Muir Wilderness/Eastern Sierra)
o Glacier Lodge (John Muir Wilderness/Eastern Sierra *)
o Vermillion Valley Resort (Muir/Adams Wilderness, Western Sierra)
o South Lake Tahoe area (multiple, Desolation Wilderness *)
o Lake Crescent (Olympic NP)
o Estes Park (multiple, Rocky Mountain NP)
o Zion Lodge (Zion NP)
* = check conditions due to recent fires
If I had to pick "one", you really can't go wrong with the Yosemite high country (Tuolumne Meadows/Tioga Pass/Lee Vining)--there's enough day hiking there to fill several days, even if you don't visit crowded Yosemite Valley (save that for fall/winter/early spring).
In the Sierra there are also several backcountry camps that you can hike into into and do day hikes from. These offer hot meals, tent cabins, beds, and (usually) showers. There are five such camps in Yosemite--all but one (Merced Lake) are usually sold out months in advance. There are two in the Sequoia area: Bearpaw also sells out months in advance, but the new Sequoia camp seems to have good availability, but is more limited in day hikes from the camp.