Can you store password in Putty sessions?

The short answer is no. But you have a workaround that can help you to reach the same result: a one-click Putty shortcut to your server from your desktop.
That's the issue of this article.

Here is how to do:

1. Create on the desktop a shortcut to Putty.exe
2. Right click on the shortcut, select "properties" and edit the target attribute
3. Change
   "C:\ProgramFiles\putty.exe"
   in
   "C:\ProgramFiles\putty.exe" root@207.150.12.233 -pw YourPassword
   where
   "C:\ProgramFiles\putty.exe" is the correct path of your putty.exe file
   "root" to the user you would use to login to your remote server
   "207.150.12.233" is the remote server IP
   "YourPassword" is the correct password.
4. Click "Apply" then "OK" to commit the change.

Now simply double-click on your new shortcut and you'll be logged on your server.