Updated June 12th, 2019.
If you can’t sign into web pages using Safari on your Mac it might be because your cookie preferences are set incorrectly. Some website sign-in pages won’t work unless cookies are enabled, so go to Safari/Preferences…, click on the Privacy button, and make sure you are NOT blocking all cookies.
(People tend to think that turning cookies off is “better” but this is not necessarily the case. The best option for you is to allow cookies from sites you visit.)
Here’s how the Privacy settings should look in macOS Mojave’s Safari.
And here’s how the Privacy settings should look in macOS El Capitan’s Safari.