It's very difficult. I like Xiao Lu (小露露) on Elizabethenstrasse opposite the McGuiness pub, but unless you go on a weekday lunchtime for the buffet it is expensive. But so is every restaurant here, really.
French side has nothing authentic. Not tried everything on the German side, but a place called Hong Kong (or HK restaurant, or something else where HK is the main title) in Lorrach wasn't bad. Decent value but not exactly authentic.
You're going to struggle. I've said this before, but it's a pity as the Asian stores in Basel sell the right ingredients to make authentic food. I can only assume the Chinese population isn't high enough to warrant the effort. Apart from Xiao Lu. Nearest place to get a real chinatown is probably Amsterdam. Quickest is London. Cheapest to get to ... not sure. Maybe some of the places in the "Vietnamese town" in Paris 13th.
Starts to make Xiao Lu look reasonable