Skype for Mac doesn't remember my password
By clicking Sign me in when Skype starts
If you are on the latest version of Skype, it supports sign-in using Skype Name or Microsoft account.

If you select, Sign me in when Skype starts it automatically stores the password and logs you in the next time. This worked fine for me on the latest version just now.

On version 7.17 (377), you get the option to select this option from Preferences