If you're using the one-click WordPress app that allows a 512MB Droplet to be used, that would most likely be the issue since MySQL requires quite a bit of RAM.
These two errors are the ones I look for within the error logs:
2017-07-05T18:31:31.155277Z 0 [ERROR] InnoDB: mmap(137428992 bytes) failed; errno 12
2017-07-05T18:31:31.155306Z 0 [ERROR] InnoDB: Cannot allocate memory for the buffer pool
Anytime you're seeing those two errors, you either need more RAM or you need to tune MySQL. In a low-RAM environment, it's not always ease to run a full stack unless you're able to tweak & tune the configuration for each individual piece of software (Apache, PHP, MySQL, etc).
The newer WordPress one-click app that's now available requires at least a 1GB Droplet (512MB is no longer an option) which helps to prevent this, though in some cases, 1GB isn't enough either.
Caching can help to reduce some of the usage (WP Super Cache, W3 Total Cache) by reducing some of the redundant database calls for each page when it hasn't changed though, so that'd be one thing I'd recommend setting up. WP Super Cache is probably the easiest to setup overall.
The thing to remember with caching plugins is that you only need one. In this specific case, more is definitely not better :-).
As with any big change -- before installing a plugin you've never used, I'd recommend taking a full backup of your site using a snapshot, that way you can fall back and redeploy to the state the Droplet was in before you installed. Caching plugins generally don't cause issues, though it's always better to be safe :-).