For a mixer site, I do not see they really need to worry about DDOS attack much. The sites do not need to handle much traffic as busy sites like this forum or some blogs or e-commerce sites
It's not used only to handle high traffic but BTC mixers are targeted with DDOS attacks perhaps more often than Bitcointalk is, especially if it's popular service. The more popular it is, the more DDOS it gets.
Between blackmail extorsions, I even suspect that some Mixers attack other Mixers, among other methods they use