You cannot take a dog to a Badi, and Werdinsel has one (and signs that dogs are not allowed). I don't recall how big the Badi's area is but I doubt it's gonna be the whole island. Perhaps the western part will be OK.
There are many places along the Limmat where you can enter the water. There are even more now, because the water is surprisingly low! For example:
a) northern / eastern side of the river to the east of Europabrücke (Kloster-Fahr-Weg),
b) or, moving more towards the city, on the same side of the river, next to Hardturmsteg, there is a small rocky "beach",
c) my favourite spot is Wipkingerpark, there are big, long concrete stairs between the path and the river. Big enough to set a BBQ or lay down on a towel to suntan. Many people come here with dogs too.
Send me a PM if you need more details. I live in that neighborhood