What you've put in prometheus.yml under 'targets' is wrong. Blackbox_exporter does not run external programs such as curl. The target should be a URL, not a command line.
I suggest you forget about prometheus.yml to start with, and get blackbox_exporter running by itself - and you can hit blackbox_exporter with curl. If you do this, then you can add debug=true to get more information:
Only after you've got that working, then you can move on to configuring prometheus.
Assuming that you can connect to the API directly using -u username:password, then what you're looking for is "basic auth":
modules:
http_2xx_with_auth:
prober: http
http:
basic_auth:
username: my_username
password: my_password
... other settings as required (e.g. method: POST, headers, body etc ...