I recently published a routing service to our server with road barriers in it and loaded it as an ArcGIS Online utility. I've added the service to the directions widget through the Web App builder, and the routing function works, but the barriers do not hold. It works fine on my desktop, but not through AGO. I am also using my own geocoding service if that makes a difference.
Thanks!
In WAB, there is not add barriers function for now. I think this is an important feature, I have submitted an enhancement in our issue repository. Thanks!