This is actual for following versions:

node 10.16.0
mocha 6.1.4
typescript 3.5.3
vscode 1.36.1

Following configuration allows to run all the mocha typescript tests or the current open test file.

{
    // Use IntelliSense to learn about possible attributes.
    // Hover to view descriptions of existing attributes.
    // For more information, visit: https://go.microsoft.com/fwlink/?linkid=830387
    "version": "0.2.0",
    "configurations": [
        {
            "type": "node",
            "request": "launch",
            "name": "[TEST] all",
            "program": "${workspaceFolder}/node_modules/mocha/bin/_mocha",
            "args": [
                "--require", "tsconfig-paths/register", 
                "--require", "ts-node/register",
                "./src/**/*.spec.ts",
                "--timeout", "999999",
                "--colors"
              ],
        },
        {
            "type": "node",
            "request": "launch",
            "name": "[TEST] current",
            "program": "${workspaceFolder}/node_modules/mocha/bin/_mocha",
            "args": [
                "--require", "tsconfig-paths/register", 
                "--require", "ts-node/register",
                "${relativeFile}",
                "--timeout", "999999",
                "--colors"
              ],
        },
    ]
}