Here is a possible solution:

Create a setting pointing to your config file (e.g. in .vscode/settings.json):

{
  "jasmineExplorer.config": "jasmine.json",
  ...
}

In the jasmine.json file, you tell the explorer where the specifications are. Here is an example:

{
    "spec_dir": "site/dist/tests",
    "spec_files": ["**/*[sS]pec.js"],
    "helpers": ["helpers/**/*.js"],
    "random": false,
    "seed": null,
    "stopSpecOnExpectationFailure": false
}

Solved by changing my test project name from DIB.MemberCatalog.Tests.JS to DIB.MemberCatalog.Tests.Client. Confusion happens when the project has the javascript file extension at the end of the project name.

You should add jasmineExplorer.nodeArgv configuration property with ["-r", "ts-node/register"] value to your settings.json (see TypeScript support section).