We can solve the given equation by taking logarithms to the base 2 on both sides. This gives:
\begin{align*}
8^{x+1} &= \frac{1}{4}\\
\Rightarrow \quad 2^{3(x+1)} &= 2^{-2}\\
\Rightarrow \quad 3(x+1) &= -2\\
\Rightarrow \quad 3x+3 &= -2\\
\Rightarrow \quad 3x &= -5\\
\Rightarrow \quad x &= -\frac{5}{3}
\end{align*}
Therefore, the value of $x$ is $-\frac{5}{3}$.
So, the correct option is \(-\frac{5}{3}\).