Standardizing just slides the variables up or down (to make their means equal to $0$), and squeezes or stretches their scales, to make the resulting SDs equal to $1$. It doesn't change the relationship between variables. However, multicollinearity is about the relationship between the variables. As a result, standardizing has no effect on multicollinearity.
If your problem with multicollinearity is due to creating product terms or interaction terms, standardizing can help if you standardize before you create the new terms. That's the only case, though. It's also possible to get collinearity with the intercept, but that doesn't matter—you can ignore that, if that's the diagnostic you're worried about.
In short, standardizing is a red herring here. It may help you to read this CV thread: When conducting multiple regression, when should you center your predictor variables & when should you standardize them?