This can be done by using closure properties of CFLs.
Let the language
$$A = \{ \text{w} | \text{w has more 0s than 1s\}}$$ be the CFL.
We can construct a TM $S$ for the given language in the following way:
1. We construct $B$ such that $B = A \, \cap L(M)$ (note that B will be a CFL)
2. Then, we can use testing emptiness of CFG on this, which we know is decidable.
3. If $S$ is empty, we reject. Else, we accept.