jmc

We use long division to find that the decimal representation of $\frac{12}{37}$ is $0.\overline{324}$. When $308$ is divided by $3$ there is a remainder of $2$ $\left(308÷3=102R2\right)$. Therefore the 308th digit to the right of the decimal point is the second digit of $324$, which is $\boxed{2}$.