/crx/login.jsp

There are curl command examples:
http://dev.day.com/docs/en/crx/current/administering/backup_and_restore.html#Automating%20Backup%20Creation?

For example,

curl -c login.txt "http://localhost:7402/crx/login.jsp?UserId=admin&Password=xyz&Workspace=crx.default"

Of course, they don't work.

You need this instead:
curl -c login.txt -F"_charset_=UTF-8" -F"UserId=admin" -F"Password=admin" -F"Workspace=crx.default" "http://localhost:7402/crx/login.jsp"

You need POST request.. and also include _charset_ param.