As far as i remember Netflix has closed its public API, and you are not able to issue keys anymore. So the only way to check if credentials are correct is to create a program that will simulate browser behavior by sending appropiate requests. You can debug netflix sign in pipeline by using for example Fiddler and than you have to simulate it in your code, checking if user was successfully logged in.
You can also write some fast and easy "automation tests" using
if you only want to check
credentials and save it to other file when test passes, it will allow you to avoid all restrictions related to browsers, as it is doing exactly the same things user manually do.