The closest distance is the perpendicular from the point to the line.
The equation of the perpendicular has slope 1/3 so the equation of the perpendicular is:
y-2=(x+6)/3 that is y=x/3+4. This meets y=-3x+4 when -3x+4=x/3+4, i.e., x=0 and y=4.
So now we work out the distance between (0,4) and (-6,2): √6²+2²=√40=6.325 approx.