What visible star is closest to the ecliptic?
Regulus ($\alpha$ Leo) is the one to beat, with ecliptic latitude $\beta$ = +0.46$^\circ$ and apparent V magnitude +1.4.
Some dimmer stars are closer to the ecliptic, e.g.
$\alpha$ Lib ($\beta$=+0.33$^\circ$, V=+2.7),
$\delta$ Gem ($\beta$=-0.18$^\circ$, V=+3.5), and
$\delta$ Cnc ($\beta$=+0.08$^\circ$, V=+3.9).
For more suggestions, poke around in Stellarium with the Ecliptic turned on.

If you want linear distance from the ecliptic plane, multiply the distance from us by $\sin \beta$.