As far as winds go, the act of mooring was relatively straightforward. The British preferred using a "high mast." The airship would approach the mast heading into the wind. Coming near the mast, the airship would lower a cable from its nose. This would be mated to a cable lowered from the mooring rig at the top of the mast, then reeled in from both ends.
The mooring rig at the top of the mast was designed to rotate to face the airship. When the cables were fully reeled in, the passengers would be able to disembark from the airship via gangways from the mast. This method only required about 10-20 people to moor an airship.
Germany used a "stub mast." This was much shorter and much more labor intensive. As with the high mast, a cable was run out from the nose and mated to one from the mast. The airship was winched down out of the air until landing ropes could be deployed, at which point some 200 men would help "ground" the airship. The tail would be attached to what was, essentially, a railcar on a track that ran around the mast, allowing the airship to weathervane into the wind as needed. Some particularly intricate masts, like the one specifically for the Hindenburg, were also on tracks that led directly into the enormous hangar, so the airship could be moored, then hangared while under control.
Passengers would disembark directly from the gondolas from a low mast.
America used both styles, with more stubs being used than high masts. The USS Patoka, seen here with the USS Shenandoah moored, used a high mast style.
The mast on the Empire State Building was a high-style mast, but would have been a nightmare to use. Indeed, only one airship ever moored to it, one owned by Goodyear. It stayed moored for just over three minutes, as it was battered by 40mph winds. One could only imagine the nightmare that mooring the Graf Zeppelin would have been, let alone disembarking her passengers.
(edit: adding some useful sources)
Sources-
"The Mooring Mast" excerpt from The Empire State Building - PDF warning
"Mooring Masts for Airships" via googlebooks
"The US Navy Airship Mooring Mast" - PDF warning
