We recommend staying at the local Discovery Parks. We liked it because it’s a bit outside of town, so we felt very safe! It has a mix of caravan parks, hotels and motel-style stays.
Check current prices and availability here.
Here are other suggestions:
Alice Springs Tourist Park: Caravan park option in town, with cabins, powered and unpowered sites. Best if you want something simple and practical close to town.
DoubleTree by Hilton Alice Springs: Luxury hotel option, good if you want a proper room, pool and easy access to town.
Diplomat Hotel Alice Springs: Motel-style stay in town, useful if you want something simple and central.