How can I add noise to a neural network input?

For what it's worth, this is my "add binomial random variate layer":

Needs["NeuralNetworks`"] (* for ScalarTimesLayer, ScalarPlusLayer *)
With[{σ = 0.1, n = 5},
 addNoiseLayer = NetGraph[
   Flatten@{
     TotalLayer[], ScalarTimesLayer[σ*1./5], 
     ScalarPlusLayer[-1*σ], TotalLayer[],
     Table[DropoutLayer[], {n}]
     },
   Flatten@{
     1 -> 2 -> 3 -> 4,
     NetPort["Input"] -> 4,

     Table[NetPort["const"] -> 4 + i -> 1, {i, n}]
     }, "Input" -> inputDim, "const" -> inputDim]]

Use NeuralNetworks`NoiseLayer

<<NeuralNetworks`
noisy = NetGraph[{
    "noise" -> NoiseLayer[{3, 90, 160}, "Distribution" -> NormalDistribution[0, 0.1]],
    "add" -> ThreadingLayer[Plus]
},
    {{"noise", NetPort["Input"]} -> "add"},
    "Input" -> NetEncoder[{"Image", {160, 90}}],
    "Output" -> NetDecoder["Image"]
]

You can find more undocumented definitions with

NeuralNetworks`$LayerData["Noise"]//Dataset

In Mathematica 12.1, DropoutLayer exposes his mask, so it's possible to train using a random binary mask, with DropoutLayer["OutputPorts" -> "BinaryMask"] (or DropoutLayer["OutputPorts" -> {"Output", "BinaryMask"}])

In Mathematica 12.2, it will be possible to simulate many types of noises (including once where you give the parameters -like mean and variance- through outputs of other subparts of a NetGraph). The novelty is already documented in the current 12.2 builds under the name of RandomArrayLayer.