If $A$ is a set of prime numbers then a necessary and sufficient condition is that $\sum_{p\in A}\frac{1}{p}=\infty$.

The general situation is more complicated. For example suppose $A$ is the union of $(n_k,(1+\eta_k)n_k)\cap \mathbb{Z}$ where $1\leq n_1<n_2<\cdots$ is a lacunary sequence. If $\sum \eta_k<\infty$ then the density of $M_A$ exists and is $<1$. If $\eta_k=1/k$, so $\sum \eta_k=\infty$, then the density exists and is $<1$.

Erdős writes it 'seems certain' that there is some threshold $\alpha\in (0,1)$ such that, if $\eta_k=k^{-\beta}$, then the density of $M_A$ is $1$ if $\beta <\alpha$ and the density is $<1$ if $\beta >\alpha$.