How to put or move downloaded files in different folders depending on file type?

I don't know of any Safari extension doing this but you may use Automator to create a folder action attached to your preferred download folder sorting files according to their extension or kind to various folders. That's even independent from the browser you usually use.

The article here outlines the workflow which sorts *.dmg and pictures:

Create the workflow In the Finder, create two folders on the Desktop. Name the first Disk Images and the second My Pictures.

Launch Automator (in your /Applications folder) and in the workflow template that appears choose Folder Action and click Choose. In the resulting window, click the Folder Action receives files and folders added to pop-up menu, select Other, and navigate to the Downloads folder inside your user folder. Click Choose. Select the Files & Folders library and drag these actions, in order, into the workflow area: Filter Finder Items, Move Finder Items, Get Specified Finder Items, Get Folder Contents, Filter Finder Items, and Move Finder Items.

In the first action (the first instance of the Filter Finder Items action), leave the first pop-up set to All. Then, configure the next row of pop-up menus to read File Extension is dmg.

In the second action, Move Finder Items, drag the Disk Images folder onto the To: Desktop pop-up menu.

In the third action, Get Specified Finder Items, click the Add button and navigate, once again, to the Downloads folder. Click Add. Click this action’s Options button and enable the Ignore this action’s input option.

Leave the Get Folder Contents action alone.

In the next Filter Finder Items action, leave the first pop-up set to All and then configure the row of pop-up menus to read Kind is Image. Finally, in the last action (Move Finder Items) drag the My Pictures folder on top of this action’s To: pop-up menu. Save your workflow and give it a name.

Also not an extention, but a separate program that goes into your control panel.... Hazel by Noodlesoft is great for this and many other uses. Not associated with them but the demo version allows one folder to monitor I think. Will be plenty good for your use case.

Automator if your friend, you can create a Folder Action and attach it to Downloads folder. There are Finder's action; Filter Finder Items which can help you filter items based on extensions, names, size etc. This way you can decide what happens to type of document. Other actions like Move, Copy, Rename etc would help you do what you need.

I would use the Run Shell Script action in Automator, you'd have more control, less chances of breaking, and probably faster. You can find the Folder Action Setup app in /System/Library/CoreServices/.