While altering layer transparency isn't built in to the class, it does give you the option of adding your own slider with the "content" object in the layers property. You'll have to code its functionality
var myWidget = new LayerList({
map: map,
layers: [{
layer: atlasLayer,
id: "Atlas layers",
content: "horizontalSlider",
subLayers: true
}],
}, "layerList");
<div id="horizontalSlider"
style="width:80%;"
name="horizontalSlider"
data-dojo-type="dijit/form/HorizontalSlider"
data-dojo-props="value:6,
minimum: -10,
maximum:10,
discreteValues:11,
intermediateChanges:true,
showButtons:false">
</div>