About this permit
Alaska Basin is in the Jedediah Smith Wilderness on the Caribou-Targhee National Forest, not in Grand Teton National Park. Both agencies say so. The Park Service's route page states that the Teton Crest Trail intersects into Caribou-Targhee National Forest at Fox Creek Pass and Alaska Basin, and that since it is Forest Service land, different regulations apply. Recreation.gov's Grand Teton permit carries a division named USFS - Alaska Basin whose description reads: this is not a park service managed camping zone, it is a National Forest Service Wilderness Area along the Teton Crest Trail where people can camp if they wish, and there is no nightly fee charged for this site.
So the practical answer is: a Grand Teton backcountry permit is not required for Alaska Basin. The zone exists in the park's system only so that a continuous Teton Crest Trail itinerary can account for that night, and no nightly fee attaches to it. If your whole trip is in Alaska Basin, you need nothing from the national park.
What the Forest Service publishes about the basin is short. It calls it a beautiful high elevation basin that holds a few lakes with amazing views, a favourite for hikers who want to do the up-and-over into Grand Teton National Park, gives Sunset Lake at 9,500 feet, and lists its restrictions as, in full, wilderness regulations apply. For the Jedediah Smith Wilderness as a whole it publishes 123,451 acres on the west slope of the Teton Range, and a food-storage line: bears roam Jedediah Smith, so store food in bear-proof containers or hang it high.
Before you plan around a specific rule, call the Caribou-Targhee Supervisor's Office at 208-557-5900 and ask whether the Jedediah Smith Wilderness needs a Forest Service permit or registration, what the group-size limit is, and what camping setbacks apply.
The route
Most people reach Alaska Basin from the park side, over Hurricane Pass out of the South Fork of Cascade Canyon, or from the Idaho side out of Teton Canyon. The Forest Service describes it as a beautiful high elevation basin that holds a few lakes with amazing views and a favourite for hikers wanting to do the up-and-over into Grand Teton National Park, and gives Sunset Lake at 9,500 feet. The basin sits in the Jedediah Smith Wilderness, which the agency describes as spanning 123,451 acres on the west slope of the Teton Range: jagged limestone peaks, deep canyons and sweeping views.
What that means practically: a Grand Teton backcountry permit is not required for Alaska Basin itself. The zone exists inside the park's permit system so that a continuous Teton Crest itinerary can account for a night spent there, and it carries no $7 nightly fee. A trip entirely within Alaska Basin needs nothing from Grand Teton.
Why it matters
Alaska Basin is the most commonly misfiled camp in the Teton Range. It appears as a selectable zone inside the Grand Teton backcountry permit, which makes it look like park land. It is not. The Park Service says so directly: the Teton Crest Trail intersects into Bridger-Teton National Forest at Teton Pass and Caribou-Targhee National Forest at Fox Creek Pass and Alaska Basin, and since it is Forest Service land, different regulations apply. Recreation.gov's own division for it is blunter still: this is not a park service managed camping zone, it is a National Forest Service Wilderness Area along the Teton Crest Trail where people can camp if they wish, and there is no nightly fee charged for this site.