Getting a custom Bézier curve function to work

f[n_, j_, u_] := PiecewiseExpand[BernsteinBasis[n, j, u]]
cv[pt_, n_, u_] := (f[n, #, u] & /@ Range[0, n]).pt

Test:

pts = {{0, 0}, {2, 4}, {4, 5}, {6, 0}};
ParametricPlot[cv[pts, Length@pts - 1, v], {v, 0, 1}, 
 Epilog -> {Red, PointSize[0.02], Point[pts], Green, Line[pts]}, 
 PlotRange -> {0, 6}]

Or

DynamicModule[{p = {{0, 0}, {2, 4}, {4, 5}, {6, 0}}},
 LocatorPane[Dynamic[p], Dynamic@
   ParametricPlot[cv[p, Length@p - 1, v], {v, 0, 1}, 
    Epilog -> {Red, PointSize[0.02], Point[p], Green, Line[p]}, 
    PlotRange -> {0, 6}], Appearance -> None]]

To plot:

bezplot[pt_, opts : OptionsPattern[Plot]] := 
 ParametricPlot[cv[pt, Length@pt - 1, x], {x, 0, 1}, 
  Epilog -> {Red, PointSize[0.02], Point[pt], Green, Line[pt]}, opts]

So,

bezplot[pts, PlotRange -> {0, 6}]

yields the first figure

I believe ...

bezierPlot1[n_, pts : {{_, _} ..}, opts : OptionsPattern[Plot]] := 
 ParametricPlot[ Sum[BernsteinBasis[n - 1, i, u] pts[[i + 1]], {i, 0, n - 1}], {u, 0, 1}, opts]

pts = {{0, 0}, {2, 1}, {4, 3}, {6, 2}};
bezierPlot1[4, pts,  Epilog -> {Green, Line@pts, PointSize[Medium], Red, Point@pts}, 
                                PlotRange -> Evaluate[{Min @@ #, Max @@ #} & /@ Transpose@pts]]

Review

Table[BernsteinBasis[n, i, u] # & /@ pts, {i, 0, n}] was the main problem.

Revising

I use the MapIndexed to fill in the second argument in BernsteinBasis[n, i, u]

 bezierPlot[n_, pts : {{_, _} ..}, opts : OptionsPattern[Plot]] :=
  ParametricPlot[
   Total@MapIndexed[
    BernsteinBasis[n, First@#2 - 1, u] #1 &, pts], {u, 0, 1}, 
     Epilog -> {Red, PointSize[0.02], Point[pts], Green, Line[pts]}, opts]

Test

pts={{0, 0}, {2, 4}, {4, 5}, {6, 0}};
bezierPlot[3, pts, PlotRange -> {{0, 8}, {0, 6}}]

Edit

bezierPlotSuper[n_, pts : {{_, _} ..}, opts : OptionsPattern[Plot]] :=
 ParametricPlot[
  Total@MapIndexed[
   BernsteinBasis[n, First@#2 - 1, u] #1 &, pts], {u, 0, 1},
    Epilog -> Join[
      Text @@@ (Thread@{Table[Style[Subscript["P", i], 14], {i, 0, n}],
        # + {.4, .2} & /@ pts}), 
      {Red, PointSize[0.01], Point[pts], Green, Line[pts]}], opts]

Test

bezierPlotSuper[3, {{2, 0}, {3, 4}, {4, 5}, {7, 8}}, 
  PlotRange -> {{-1, 11}, {-1,11}}]