SQL Developer should support fetching SSH keys from a running SSH agent (ssh-agent or Putty's Pageant), not only specifying a key file and asking for its password.
SQL Developer supports connections through a SSH tunnel, which is very convenient. When setting up such a connection, a key file is selected. This key is usually password protected, and therefore SQL Developer asks for the password every time it connects. Storing this password is not an option, security-wise. That's why we have SSH Agent, a program that's part of both OpenSSH and the Putty suite. This agent allow you to enter the password less frequently, and other programs use the agent instead of accessing the key file directly.