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Using RandomPoint[] with TubeMesh[] (routines from herehere and herehere) does the job:

helix = First[Cases[Normal[ParametricPlot3D[{6 Cos[t], 6 Sin[t], t}, {t, -2 π, 4 π},
                                           MaxRecursion -> 1, PlotPoints -> 75]],
                   Line[l_] :> l, ∞]];

tube = TubeMesh[helix, 1/2, "CapForm" -> "Round"];
BlockRandom[SeedRandom[42]; 
Graphics3D[{AbsolutePointSize[1], Point[RandomPoint[tube, 5000]]}]]

random points around helix

Using RandomPoint[] with TubeMesh[] (routines from here and here) does the job:

helix = First[Cases[Normal[ParametricPlot3D[{6 Cos[t], 6 Sin[t], t}, {t, -2 π, 4 π},
                                           MaxRecursion -> 1, PlotPoints -> 75]],
                   Line[l_] :> l, ∞]];

tube = TubeMesh[helix, 1/2, "CapForm" -> "Round"];
BlockRandom[SeedRandom[42]; 
Graphics3D[{AbsolutePointSize[1], Point[RandomPoint[tube, 5000]]}]]

random points around helix

Using RandomPoint[] with TubeMesh[] (routines from here and here) does the job:

helix = First[Cases[Normal[ParametricPlot3D[{6 Cos[t], 6 Sin[t], t}, {t, -2 π, 4 π},
                                           MaxRecursion -> 1, PlotPoints -> 75]],
                   Line[l_] :> l, ∞]];

tube = TubeMesh[helix, 1/2, "CapForm" -> "Round"];
BlockRandom[SeedRandom[42]; 
Graphics3D[{AbsolutePointSize[1], Point[RandomPoint[tube, 5000]]}]]

random points around helix

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Using RandomPoint[] with TubeMesh[] (routines from here and here) does the job:

helix = First[Cases[Normal[ParametricPlot3D[{6 Cos[t], 6 Sin[t], t}, {t, -2 π, 4 π},
                                           MaxRecursion -> 1, PlotPoints -> 75]],
                   Line[l_] :> l, ∞]];

tube = TubeMesh[helix, 1/2, "CapForm" -> "Round"];
BlockRandom[SeedRandom[42]; 
Graphics3D[{AbsolutePointSize[1], Point[RandomPoint[tube, 5000]]}]]

random points around helix