The original function \( f(x) = \sin x \) has a minimum value of \(-1\) since the sine function varies between \(-1\) and \(1\).
When the function is translated upward by 6 units, the new function becomes:
\[ f(x) = \sin x + 6 \]
To find the minimum value of this new function, we add 6 to the minimum value of the original sine function:
\[ \text{Minimum value of } f(x) = -1 + 6 = 5 \]
Thus, the minimum \( y \)-value of the translated function is \( \boxed{5} \).