Ok, so I spent a fair amount of time searching to find this information so I thought I may as well share it.
The best gauge near Basel is at Rheinhalle: http://www.hydrodaten.admin.ch/en/22...aktuelle_daten
This is above the Birsfelden dam, so it may not be reflective of water levels in Basel for swimming/paddling. However, it provides water flows in-addition to water level. Note that the Swiss decided to report water level as meters above sea level, which is very helpful considering how close Switzerland is to the sea. If you click the "Station Data" tap you will see the reference height of the gauge, you have to subtract that value from the graphs to get the actual water level. Unfortunately they didn't provide a red-line or other guide for the eye on the graphs so it's relatively tricky to guess the water velocity.
The gauge in Basel-proper has just water level: http://www.hydrodaten.admin.ch/en/26...aktuelle_daten
and the water temperature (and quality) is availabe from the Weil station on the German border: http://www.hydrodaten.admin.ch/en/2613.html
For completeness, here is the Birs, which is a little class 2- whitewater (wildwasser) river that dumps into the Rhein. http://www.hydrodaten.admin.ch/en/2106.html
Note that the Birs is polluted and not suitable for swimming. It has some little drops and hydraulics as it nears the Rhein, and when it's in flood it has some fairly beefy surf waves.
I haven't found anything for Huningue yet.