I followed this guide to setup SSH public key authentication.

First copy your public SSH key to the destination host

# The destination user
export USER=pi
# The destination host
export HOST=192.168.0.13
ssh-copy-id $USER:$HOST

Then test if you can now connect to the destination host without needing to enter a password

ssh $USER@$HOST

If this works, you can safely edit the destination hosts’ SSH configuration file to disable password authentication as a whole

sudo vi /etc/ssh_sshd_config

Search for the line containing PasswordAuthentication and change this line to

PasswordAuthentication no