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That isn’t too much of a load. A 1 GB droplet really should be able to handle that. Are you seeing any error messages when it crashes? Is there anything interesting in <code>/var/log/messages</code> ? We can give some general advice, but getting to the bottom of this exact issue would probably be best.
More generally, there is a chance MySQL is running out of memory. In order to give it some extra breathing room, you might want to add a swap file:
https://www.digitalocean.com/community/tutorials/how-to-add-swap-on-centos-6
You may also consider moving to Nginx as it is more lightweight than Apache. Check out:
Setting up caching with something like Varnish is also a good step:
Less specific to your issue, one way to scale out is to add a separate database server: