What is this string on the 737?
The ground crew has connected a headset to the communication jack outsiede for flight pushback communication with the flight crew. from b737.org.uk:

The External Power Hatch is located beneath the F/O's DV window. It is used by groundcrew to connect the Ground Power Unit and headset for pushback communications with Flight Interphone.
The service interphone is used by engineers to communicate with the service interphone stations inside the aircraft.

Here is a closeup of the connection point

Image from b737.org.uk
About flight interphone System

The flight interphone system is an independent communication network. Its
primary purpose is to provide private communication between flight deck
crewmembers without intrusion from the service interphone system. The ground
crew may also use the flight interphone through a jack at the external power
receptacle.

And here is a nose section of 737-200, showing it from a not so close angle.

Image from rigger.smugmug.com