It's tempting. Google, Firefox, Apple and Microsoft make it very appealing. It's just one click to save you time. After you enter your username and password into a website, the browser inevitably prompts you to "Save your Password". Then, the next time you visit the site your credentials will be ready and waiting for you! You don't even have to type them in.
DON'T DO IT.
At least on ESU classroom, lab or public access computers.
In classrooms on campus, users are encouraged to log in using the 'smart' account. This ensures that all tech in the room is configured and ready to go. The downside is that 'smart' is a shared resource now. Each user of that computer can save files to the desktop, reconfigure icons or yes--save their passwords in the browser! This means that each subsequent user--faculty, staff, student or guest, would then have access to the originators files, email, Canvas or more.
Similarly, on other shared devices--for example, checkout computers and some lab PC's, what may seem like a convenience is only giving the next student free reign into your content if you've clicked the ever-tempting "Save my Password" button.
Now, I'm sure you're thinking "but I log in with my own account to my computer and I don't share that with anyone". And you are correct, that is safer. To be completely safe, especially for credentials to highly sensitive information, just ignore that tempting prompt and click "No."