To find the coordinates of the point symmetric to the origin, we use the property that if a point has coordinates $(x, y)$, then its symmetric point with respect to the origin has coordinates $(-x, -y)$.

Given the point $(6,1)$, applying the property:

- For the $x$-coordinate: $6$ becomes $-6$
- For the $y$-coordinate: $1$ becomes $-1$

Therefore, the coordinates of the point symmetric to the origin are $\boxed{(-6,-1)}$.