Thanks, @alesh. Today I installed a fresh version of Laravel 5.6.24 using PHP 7.2 FastCGI on a Dreamhost domain. I haven't set up a database yet, but for a static website, it works.
A couple notes for others who may come here:
You can use git if you've enabled shell access. Just remove the vendor folder from your git ignore file.
PHP 7.2 is available on Dreamhost now.
I did not run any artisan commands - I just edited the .env file directly by renaming the .env.example file to .env and copying the APP_KEY from my local install.
For anyone else who may have missed the part about the /public folder, @alesh did mention it, but perhaps not in the big bold letters it deserves:
Under Web directory, add /public to the end of the domain name.