I work for a very mature startup, and recently led our first site on ASP.NET Core. The framework itself is very nice and very solid, basically a "do-over" of all the previous ASP.NET frameworks with the benefit of hindsight. Although there's been some new things to learn, I've been very pleased with how consistent and thought out the new ASP.NET Core is. We run the site in production.

Having said that, the tooling around ASP.NET Core has been an absolute disaster. They went 90% of the way with one mechanism (project.json), then decided to start over with a new mechanism (msbuild). While I believe it is ultimately for the best, the interim has been incredibly difficult to work with, as ASP.NET has been officially released now for ~8 months with neither method fully working. For example, in the current project.json stuff you cannot reference another project without first putting it into a Nuget pacakge.

Hopefully this will all be cleared up with the upcoming release of VS2017, but if you need something sooner and aren't interested in dealing with a lot of pain, I'd stick to ASP.NET classic for now.