What visible star is closest to the ecliptic?

HIP 76880 = $kappa$ Librae (V=4.72) has an Ecliptic latitude of -0.019 degrees.

This is the winner amongst all stars in the Hipparco/Tycho catalogue with a Hipparcos magnitude <6.

If you mean physical distance, rather than angular distance, then the sine of the ecliptic latitude must be multiplied by a distance estimate for the star. This cannot be conclusive, the distances to many naked eye stars are very uncertain, a small fraction of Hipparcos parallaxes are too uncertain to be useful in this regard.
However, from those with decent parallaxes then HIP 3765 = HD 4628 (V=5.72) has an absolute distance of 0.036 pc above the ecliptic plane.