How can I color a BezierCurve from Red into Green just as Line does?
pts={{0,0}, {1,1}, {2,0}}; Graphics[{Thick, BezierCurve[pts], Line[{{0,0}, {2,0}}, VertexColors->{Red, Green}]}] 
A simple line strip should be sufficient for most purposes and there you can use VertexColors as usual:
pts = {{0, 0}, {1, 1}, {2, 0}}; Graphics[{Thick, Line[BezierFunction[pts] /@ #, VertexColors -> (Blend[{Red, Green}, #] & /@ #)] &[Range[0, 1, .01]], Line[{{0, 0}, {2, 0}}, VertexColors -> {Red, Green}]} ] 
pts = {{0, 0}, {1, 1}, {2, 0}}; f = BezierFunction[pts]; Show[ ParametricPlot[f[t], {t, 0, 2}, ColorFunction -> Function[{x}, RGBColor[1 - x, x, 0]]], Graphics[{Thick,Line[{{0, 0}, {2, 0}}, VertexColors -> {Red, Green}]}], Axes -> None ] 