Outside of the "East/Southeast Asia" that one thinks of when one says "Asia," dairy is actually very large part of Mongolian, Siberian, and Kazakh diets. Airag is the national drink of Mongolia, also known as Kumiss in the Russian-speaking steppe nations. It's composed of fermented mare's (horse) milk, and becomes a lightly effervescent and alcoholic drink. Mongols also produce Aaruul in great quantities, and it is available everywhere. The sun-dried curds allow for long term storage and preservation. Other milk products include tarag, a yogurt, and urum, a essentially the milk fats from the top of fresh milk.
Also, "Food in China: A Cultural and Historical Inquiry" by Frederick J. Simoons addresses your question - Why haven't populations in China (and in other parts of south/east Asia) used dairy? The immediate counterpoint is that most of the Indian subcontinent also uses dairy (e.g. Paneer), and the foodstuffs I mention above are used by the Manchus, who today reside in China (admittedly they are closer, culturally, to the Mongolians and other steppe nomads). As for why say, Japan or Korea never invested in dairy, I've seen explanations fall into these lines of reasoning:
As shown by the steppe cultures, it's not especially difficult to produce and keep dairy products, and integrate it into your diets. As shown by India and the rest of the subcontinent, hot temperatures are not a barrier either. A combination of these factors is a likely explanation, though I can't personally say what the main drivers were. Lactose intolerance combined with ready alternatives to the nutrients via leafy green vegetables seems to have led to the lack of dairy, at least to me. My main point, however, is that many Asian cultures have traditionally produced and used a lot of dairy.
And on a side note, could the absence of this western staple in Asian cultures be a reason many Asian people are more diminutive when compared to western people?