I know that many of us that run systems for which availability criteria mean they want full manual control over all updates, but those are the people that either use custom images or know how to turn those automatic updates off.
Security updates are finally pushed hard on desktop users for obvious reasons, with everyone and their mother running cheap VPS's these days the same logic should apply to virtual servers.
Ubuntu, for example, can be configured to run apt-get update && apt-get upgrade automatically every day (using unattended-upgrades), but AFAIK the default configuration only checks for updates, notifies the admin, and waits for approval before actually changing anything.
Even though Ubuntu LTS, Debian Stable, CentOS, etc. are supposed to be stable distributions, it is not unheard of for routine updates to break something mission-critical. For example, the package maintainer might have made some changes to the default configuration. (C'mon, do you really need to make minor changes to nginx.conf all the time?) A daemon might go down and fail to come up again after an update. (apt is particularly annoying in this regard, since it forcibly restarts daemons during an upgrade, and not always in a sensible order. Result: HTTP 502 Bad Gateway.) So unless a human is present to spot any issues immediately after an update, you might be left with a broken system at an inconvenient hour. Which is why the documentation doesn't recommend fully automated updates.
Unfortunately, the set of VPS owners who don't apply updates regularly probably has a large intersection with the set of VPS owners who won't be able to troubleshoot a broken update anyway, so maybe this concern need not apply...
(CoreOS employee)
I don't think DO offers any managed service. They wouldn't do anything to fix your servers. It's entirely on you to keep them updated.
Edit: https://www.digitalocean.com/help/policy/ <- they really don't offer it