In Southeast Asia, civilization is not crippled; a large part of Southeast Asia is maritime, and as long as there are people building boats and sailing, civilization survives. The location of the centers of civilization will be different, however, especially on the mainland; instead of capital locations like Burma's Pagan, which were built in or near prime horse breeding territory, harbor cities will become more prominent early on. On land, elephants will become even more important militarily than OTL, as there will not be a reliable cavalry in the region due to the diseases spread by tsetse flies. China will also suffer without southeast Asia's horse breeding to supply their imperial cavalry during times when relations with the steppe nomads go south, so China will probably be more regularly overrun by invaders from the north ITTL.
The diseases spread by tsetse flies are, IIRC, not as dangerous to donkeys so it is possible that ITTL European powers will have more luck introducing the practice of breeding donkeys with horses and creating mules than they did IOTL.
This article has some interesting information on horses, donkeys and mules in southeast Asia.