The phrase “Surfing the Internet” seems, based on some modest research, to have originated in the early 1990s. One person credited with its popularization is Jean Armour Polly, a librarian who used the metaphor in an article published in 1992. From Surfing the INTERNET: an Introduction:

Today I’ll travel to Minnesota, Texas, California, Cleveland, New Zealand,
Sweden, and England. I’m not frantically packing, and I won’t pick up
any frequent flyer mileage. In fact, I’m sipping cocoa at my Macintosh.
My trips will be electronic, using the computer on my desk, communications
software, a modem, and a standard phone line.

I’ll be using the Internet, the global network of computers and their
interconnections, which lets me skip like a stone across oceans and
continents and control computers at remote sites. I haven’t “visited”
Antarctica yet, but it is only a matter of time before a host computer
becomes available there!

The totally sweet art in the header-header is cropped from the English-language cover of the book Das Internet – Surfen im Computernetz by Christof Hafkemeyer with illustrations by Joachim Knappe:

Surf it, live it, love it.