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By default MySQL does not listen on your public IP address. This can be changed in the /etc/mysql/my.cnf file on your droplet.
Commenting out or changing the line below to 0.0.0.0 (all IPs) or your public IP address
bind-address = 127.0.0.1
and then restarting with
service mysql restart
should correct this and allow your remote client to connect.