I admit to writing them down too ... plus my iMac, iPhone and iPad sync to share the info - so what ever device I am using, I just use the Autofill feature. However that feature does not always work properly - so I do indeed write them down and keep them in a page protector. In fact I just updated my very long list at the beginning of the month so it's current. I have almost a full page (front and back of typed websites & corresponding passwords!) It's unreal how many passwords we need to have in today's world. I cannot wait until there is a better way.
Ugh! I have so many passwords that I no longer can remember them all. I let my browser remember the ones that don't really matter, but the worst is my bank that makes me change my password every 3 months. There is no way I'm going to remember all those passwords, so I have to keep track of those. sigh.
I use Last Pass which is cloud based encrypted storage for my passwords. It will generate passwords and automatically updates itself when you change one. You remember your Master Password and then it syncs to my desktop, laptop, volunteer computer, and phone/iPad. I love it. https://www.lastpass.com/
I store mine old school. I keep a log book and write them down. I don't keep them digitally anywhere. At our office (I'm an IT Administrator), we use 1Password, which is a software program you can purchase for password management.
I heard a clever idea the other day. Use your standard password and the first letter of the site, like L for the Lilypad