To accomplish this task in Adobe Illustrator, use Effect > Distort & Transform > Roughen. Set the Points to "Smooth" and adjust the Size and Detail downward.

Convert the stroke to outlines with Object > Path > Outline Stroke, then use the Bloat and Pucker tools (hidden under the Warp tool) to adjust the stroke width. To adjust the properties of the tools, select the tool, then double-click on its icon in the toolbar. Or you can hold down alt to interactively change the brush size prior to using it.