If you write the coordinates of a point as $(x_1,x_2,x_3)$, then the square of its distance from the origin using the given metric is $$\sum_{\lambda=0}^3\sum_{\mu=0}^3 g_{\lambda\mu} x_\lambda x_\mu = 2(x_1^2+x_1x_2+x_2^2+x_2x_3+x_3^3).†$$ Plug in the coordinates of an arbitrary point on the line to get the distance squared as a function of $t$ and minimize (and don’t forget to take the square root at the end).
You can also do this without using calculus. The line can be expressed as $(-7,4,4)^T+t(3,-2,3)^T$, so if you translate by $(7,-4,-4)^T$ so that the line passes through the origin, the problem is equivalent to finding the distance from this point to the translated line. This is a matter of computing the orthogonal rejection of $\mathbf p$ from the direction vector $\mathbf v=(3,-2,3)^T$ of the line, using the inner product induced by $g$: $$\mathbf p_\perp = \mathbf p - {\mathbf p^T G \mathbf v \over \mathbf v^T G \mathbf v}\mathbf v$$ and the distance is then $$\sqrt{\mathbf p_\perp^T G \mathbf p_\perp}.$$ Here, $G$ represents the matrix with entries given by the components $g_{\lambda\mu}$ of the metric.
† You might also see this written more simply as $g_{\lambda\mu}x^\lambda x^\mu$, where the coordinates use superscripts instead of subscripts and the Einstein summation convention is in force.
