Does Terraria on Steam need Internet connection after download?

No. It does not need any connection if you play only singleplayer. :)

No it does not need Internet connection after it has been downloaded. There is a "Single Player" and "Multiplayer" option on the menu, clicking the Single Player button will create a world that is stored on your computer and can be played completely offline.

To connect to a server using the "Multiplayer" menu, you will have to be connected to the Internet. You will also need to be connected to the Internet if you want to use the Steam Cloud feature to have your characters and worlds be accessible from any computer that you are logged into Steam on.