We're using Stripe for an ecomm project that we're testing out right now and nothing is hitting Stripe. If we use a bad card number (per Stripe's docs) like 4000000000000002, the order still goes through.
The Stripe account is brand new (hasn't fully been activated yet) and in Test mode and the test secret and publishable keys are in Commerce's settings. I'm not sure what else to check.
EDIT:
I had forgotten stripe.js altogether but now I'm getting "Payment information submitted is invalid." which I think is a Craft/Commerce error. Here's a link to my checkout template code. The wrapper template (extended template) contains this javascript before the closing body tag.
stripeResponseHandler
in your JS? So basically your code is hitting Stripe with the card details to get the token. So you have the token, now your JS needs to insert that token back into the form so Commerce on the server side can actually submit the form and do what it needs to do with the token.