Prepare to be surprised....
All ships out there are still berthed/unberthed manually using the experience and judgement of the Master, Pilot (if carried) and Officers onboard - there is no automation involved at all save for natural modernisation of control systems e.g. no more clanking polished brass telegraphs with their bells and chains for engine orders and wooden wheels to steer by, but electronic switches and levers.
Even those vessels fitted with dynamic positioning equipment (offshore oil industry) are still manoeuvred manually when in close proximity to hazards or structures.
I've no doubt systems exist where it might be possible to bring some form of automation into play using sensors/cameras etc and the "machine" does it all, however I've no knowledge of any vessel where that is the case - although saying that there are probably one or two out there in highly specialised and controlled circumstances.
You may be equally surprised to learn that even today there is no mandatory training in shiphandling available to ship's officers nor are they examined on it by their flag state. There are naturally unofficial courses you can attend to get a gist of the basics using theory and limited simulation, however it's still primarily learnt "on the job" both under supervision and without.
P.S. The modernisation of control systems mentioned earlier basically means those on the bridge have full control over engine movements, i.e. they have the levers and dials up there, as opposed to past times where you rang down your engine order on the bridge telegraph. This was a large, highly polished brass piece of kit (usually manufactured by someone like Chadburns) with a dial and levers and which was connected by a chain all the way below to an engine room repeater which displayed the order transmitted on a dial, once received the engineer below would repeat the order upwards using his lever which would move a small arrow on the bridge telegraph thereby assuring those therein that their order had been correctly received. Naturally this always led to a very slight delay between asking for an engine order and getting the desired response.
This system was however useful when things all went wrong and the vessel either landed heavily on a quay or collided with something, as the officers on the bridge would usually place fault on those in the engine room by alleging that they hadn't answered engine orders quickly enough or indeed correctly. With modern systems which are controlled by those on the bridge, that get out of jail free card potential incident root cause is no longer usually applicable.