On clicking "Launch In Docker' I get a dialog with the following message:

The folder Docker-TypeScript is ignored since it uses task version 0.1.0

I used the instructions here to convert the tasks.json file to the following:

{
  "version": "2.0.0",
  "tasks": [
    {
      "label": "tsc-watch",
      "command": "npm",
      "args": [ "run", "watch" ],
      "type":"shell",
      "isBackground": true,
      "group":"build",
      "problemMatcher": "$tsc-watch",
      "presentation":{
        "reveal": "always",
      }
    }
  ]
}

This appears to fix the immediate issue. (My breakpoints aren't working but this may be another issue).

My environment:
Version: 1.31.1 (user setup)
Commit: 1b8e8302e405050205e69b59abb3559592bb9e60
Date: 2019-02-12T02:20:54.427Z
Electron: 3.1.2
Chrome: 66.0.3359.181
Node.js: 10.2.0
V8: 6.6.346.32
OS: Windows_NT x64 10.0.17134