Both award and Basic Economy (now) allow a full size carryon bag that fits in the overhead bin and a personal item that fits under the seat in front of you. But practically speaking, Basic Economy generally is Boarding Group 9 and the likelihood is strong the carryon will Be Gate Checked to your final destination, as it’s likely the bins will be full. (Of course, this is possible on nearly all bookings, including award bookings, if one is boarding in one of the last groups.)