There's a lot of words here but trust me, this is pretty simple to get running. It'll take you less than 5 minutes. Might even take you less than 1.

First off, if you haven't already, activate controller support with Steam Big Picture. Go into Steam Big Picture (the square icon at the top right of Steam with the arrows pointing), go onto the cog icon top right and then go to controller settings. If you have an Xbox controller, click on the Xbox controller support and then exit. If you have any other, I assume the option will appear there. You don't need to use Big Picture again. This is a one-time setup.

First, download and install T2Fix if you don't have it already. It's basically a necessity anyway if you want a smooth experience playing this game, with working cutscenes, widescreen support, bug fixes and so much more.

Set your controls in-game to ‘Thief 1 Default Bindings’ by pressing ‘Load’ in the ‘Customize Controls’ and selecting ‘Thief 1 Default Bindings’.

Note: If you can't find these bindings, you may not have T2Fix installed.

Some controls aren’t bound by default, so you’ll have to change them.
Change the ‘Prev Weapon’ keybind to ‘Numpad .’, also known as ‘Numpad Delete’.
Change ‘Prev Inventory Item’ keybind to the letter ‘k’.
Change ‘Next Inventory Item’ keybind to the letter ‘l’.
Change ‘Lean Left’ keybind to the letter ‘v’.

Apply this configuration: steam://controllerconfig/211740/2074875863

Alternatively, in Steam Big Picture you can find my configuration under 'Thief II Controller Bindings' in the community tab of exported controls.

Yay. Controller support enabled. Enjoy!

The controls were made to be as comfortable as possible, but given how this isn't truly native, some things have to be a bit odd. I'm sure you'll get used to it very quickly, however.

The controls are as follows:
Left trigger/LT = Use
Left bumper/LB = Lean left
Back button = Quicksave
Left stick movement = Move (Move fully in any direction to run, move slightly in any direction to walk. Full pressure is needed to run, so you likely won't accidentally run.)
Left stick click/LS = Crouch
Up Directional Pad = Lean forward
Left Directional Pad = Weapon scroll down
Right Directional Pad = Weapon scroll up
Down Directional Pad = Drop item (Quieter than throwing.)
Y/Yellow = Put away item
X/Blue = Item scroll down
B/Red = Item scroll up
A/Green = Jump
Right trigger/RT = Attack
Right bumper/RB = Lean right
Start button = Pause
Right stick movement = Camera move
Right stick click/RS = Put away weapon
Xbox guide button = Unused

Unfortunately I couldn't fit in any buttons for zooming, so you'll have to use the keyboard for that.