You could login as the root user on the droplet. After this create a new user (on CentOS: useradd username). After you've created the user, login as this user (can be done directly from root: su username).
Now edit the file: ~/.ssh/authorized_keys and paste the public key in this file. Save it and make sure you chmod it to 600. Now you could login as the newly created user with the SSH key.
If you're unable to login, please check the /var/log/secure login (as the root user) to see why you can't login. If it says something about not in the AllowUsers, then: nano /etc/ssh/sshd_config and at the end of the file, you could add the user: AllowUser username. Make sure to restart ssh.