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
}

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