When The Website Is Abandoned?
There is very little you can do here, although sometimes encouraging the author/owner can revive the stagnation. Just knowing that one has a caring reader is often enough to spur on new efforts.
When The Website Is Down?
Before you send a nasty email to the owner about how important your fix is, perhaps you should make sure it’s not just you.
Is It Just You?
First and foremost, figure out if it’s just you with Down For Everyone Or Just Me?. If it’s just you, make sure your internet is working, refresh the page, restart your browser, try clearing cookies and caches. You can also do some tinkering like pinging, tracert, proxies, etc.
The Website Is Down!
Your options become limited. The most obvious is to contact the owner. The less obvious thing to do is figure out why. Check whois records to determine if the domain expired. Check network/site activity charts from third parties to determine how long the outage has lasted. If it’s an outage, chances are that the site will restore itself over time (the webhost is probably at fault here and they work behind the scenes to clear the problem).
There’s An Error That Makes The Website Useless
Your only option here is to try to find a workaround to accomplish what you are trying to do and tell the owner about it.
The Domain Expired
Most webhosts and domain registrars have policies in place never to share account details with anyone except the account holder – this also means that you cannot donate your own money directly to the hosting/domain company.
You can try to gather up any needed materials with The Internet Archive or even Google cached search results pages (click the ‘cached’ to view the cached page instead of visiting the dead website). If you can gather up enough old webpages, you may want to consider mirroring the dead site (creating a new website with the old website’s materials). For example, this is how Bonsai Kittens meme survives on the web, thanks to a generous fan.