Browser Twine is updated whenver there's a new release, so there's no work for you to do to stay up-to-date. If you need to use an older version, go to https://twinery.org/[version number]. That is, to use 2.3.13, go to https://twinery.org/2.3.13. These different versions will use the same story library and preferences.
Keep in mind, though, that earlier versions will have bugs the most recent version does not, and there might be strange behavior moving backwards in time, so you should only use an older version if you really need to.
To check if a newer version of Twine is available, choose Check for Updates from the menu named Twine in the menu bar (not the top toolbar). Twine will tell you if there's an update available and, if you like, will direct you to where to go to download the new version. Twine doesn't automatically update itself.