The integration appears to be working fine for us now. When clicking the "log into GTW" from within Admin and a blank window pops up/disappears, that could be an indication that the connection is fine. If not - let's say the password changed - they you would have an opportunity to provide updated login credentials.
I, too, have been having this same issue since 3/14. It caused a huge problem with a recent webinar and we were unaware of the issue because Marketo never provided a notification that the API connection was broken. We've since tried to re-establish the connection and continually fails. I'm almost glad to see others are experiencing this -- we were going crazy trying to troubleshoot.
I am going to assume you've fixed it by now - but for anyone else having the same issue (I just had it today, Oct. 24) take these steps to correct it.
When I would click LOGIN TO GTW - it would look like it was processing, then return to the same error (API/403) under details. Combining things above - I simply:
1. Opened an incognito window
2. Logged into Marketo
3. Logged into GTW from Launchpoint
Closed the window and the error was gone - I returned to the non-incognito window, hit refresh and it was resolved there as well.
I suspect this was due to someone changing the password on the account.