A work-around that works with arbitrary number of axes: post-process to inject the option AxisLabel to AxisObjects:
ClearAll[fixMultiAxisLabels] fixMultiAxisLabels[off_ : 1.5, opts : OptionsPattern[]] := Module[{al = #2}, ReplaceAll[Offset[o : Except[0, _], z_] :> Offset[off o, z]]@* ReplaceAll[AxisObject[a_, b_, c___] :> AxisObject[a, b, Last[al = RotateLeft[al]], opts, c]]@#] &;
Examples:
llp = ListLinePlot[{{1, 2, 3}, 10 {3, 2, 1}}, MultiaxisArrangement -> All, ImageSize -> 500]; labels = {"A", "B", "C", "D"}; axislabels = Thread[AxisLabel -> Placed @@@ Transpose[{labels, {0.5, {0.5, #}}&/@ {-2.5, 2.5, 2.5, -.5}}]] fixMultiAxisLabels[][llp, axislabels]

llp2 = ListLinePlot[{10000 {1, 2, 3}, 10 10000 {3, 2, 1}}, MultiaxisArrangement -> All, ImageSize -> 500, LabelStyle -> 16]; axislabels2 = Thread[AxisLabel -> Placed @@@ Transpose[{labels, {0.5, {0.5, #}} & /@ {-3.5, 5, 2.5, -.5}}]] fixMultiAxisLabels[][llp2, axislabels2]

An example with 4 vertical axes:
llp3 = ListLinePlot[{{1, 2, 3}, 10 {3, 2, 1}, 500 {2, 1, 3}, 1000 {1, 3, 2}}, MultiaxisArrangement -> All, ImageSize -> 800]; labels3 = {"left 1", "left 2", "right 1", "right 2", "bottom", "top"}; axislabels3 = Thread[AxisLabel -> Placed @@@ Transpose[{labels3, {0.5, {0.5, #}} & /@ {-2, -1.25, 3.25, 4, 2.5, -.5} }]]; fixMultiAxisLabels[2, LabelStyle -> 14][llp3, axislabels3]

InputFormof that plot does not look right. Seems like a bug to me. You should report it to Wolfram Support. $\endgroup$MultiaxisArrangementis used:PlotLegendsis ignored. $\endgroup$