2 hotels are located right next to Almaty airport: Ekipage and Aksunkar. Ekipage is not great, Aksunkar is awful. A bit further off, 2 better, cheaper options exist: Kuze Hotel and Altyn Kol Hotel. For more hotels, see the Almaty city guide.
Kuze Hotel offers basic rooms, basic service and basic breakfast for a small price (by Almaty standards). It is close to Almaty-1 train station, though this is not the best neighborhood in Almaty. It is clean, though, and there is free wifi.
Altyn Kol has lovely wooden houses in a pretty setting on the lakeside. Staff is very kind and helpful. This is a nice rustic-style getaway with a Kazakh-Siberian atmosphere, but beware: weekends are reserved for noisy wedding parties and the breakfast only starts at 10!
Ekipage Hotel: For 100$ per night, one expects a bit more than this. Basic rooms, unfriendly staff and a shitty breakfast. It is also very noisy due to the landing planes. Pick up from the airport is free but drop off is a ridiculous 2000 tenge. But, if all you want is a place to stay for the night, really close to the airport, Ekipage is your best bet.
Aksunkar is the actual airport hotel on the airport terminal premises. Prices start from 85$, quite a price for a dingy, cold, dirty room with uncaring, Russian-only staff. You can do better!