While any of several SMS service providers should do the trick, it's worth nothing that Digital Ocean posted a guest blog post for Twilio a while back, talking about how Twilio and Digital Ocean play well with each other: https://www.twilio.com/blog/2014/10/digitalocean-communicates-clearly-with-twilio-sms-nt.html
Personally I've had good experiences with using IFTTT's "Maker Webhooks" plugin. It allows me to send a curl command (in good old command-line Bash) from a droplet to maker.ifttt.com to trigger a generic IFTTT "event". The event can then be translated into sending an SMS message, or sending an email, or turning on a Philips Hue light, or an LED or my Raspberry Pi, or whatever.
I haven't used Twilio in several years, but back then at least the experience was also quite good. It costs money, but not much, and seemed very reliable and easy to use.