I do not believe the growth of the Internet has been the cause of the decline of the BBS. On the contrary, Internet technology could have boosted the world of the BBS. Unfortunately, the BBS world was in serious decline since about 1990, several years before the Internet explosion began.
I used to run Astral Board a popular BBS in Pittsburgh, PA in the 1980's and early 1990's. I started it with a 1200 bps modem, a Tandy 1000 computer and the Collie BBS software. That despite the fact that all the other BBS's in Pittsburgh used Fido, which, somehow, I never cared for.
I later switched to Opus, even created the Avatar protocol for it (which allowed screen control much faster and more powerful than ANSI sequences). I actually took over the development of Opus from the original author for a while, and postponed becoming a US citizen for a year, so busy was I working on Opus.
Originally, each BBS was unique. It reflected the interests of each individual sysop. Most major US cities had several BBS, each quite different. Their sysops were usually quite friendly with each other. Well, at least in Pittsburgh we were, I cannot speak for other communities.
This gave BBS users a choice. Ironically, most chose to visit most BBS's, not just one or two.
Most BBS's were members of Fidonet, which allowed them to exchange messages, just as in present-day email, except they usually exchanged them for only one hour at night when long distance charges were lower: It was the sysop who had to pay for it (and, for that reason, many did not allow it, or only used it for themselves).
Then, someone came up with the idea of echos. A typical pre-echo BBS had several different message areas, each dedicated to a specific topic. Of course, there was some overlap: Different BBS's may have had message areas dedicated to the same topic. The idea behind the echo was that friendly sysops could exchange the messages of compatible message areas.
This was great, especially since it allowed users of a message area in one city to discuss their ideas with users from another city, soon to be several cities, soon to be all over the world.
Through various collaborative efforts it became possible for the various BBS's to participate in an unlimited number of echos at minimal or no cost to the sysop.
Then something strange happened. Many new BBS's started popping up. To attract users, their sysops started participating in almost every echo there existed. Suddenly, the traditional sysops started losing users to these new all-echo message boards. To keep up, most BBS's started carrying most echos.
All BBS's suddenly were the same. They lost their individuality. Users no longer called all BBS's in town. What for? Just call one, about once a day, and you have seen it all.
Furthermore, someone came up with the idea, and the software, that it was no longer necessary to log into a BBS to read messages and reply to them. Instead, the software logged in automatically, pretending to be the user, and downloaded all new messages in the echos the user was interested in.
The user now could read the messages offline. The user could also compose replies offline, and just have the same software "upload" the replies back to the BBS.
The typical BBS suddenly became nothing more than a mail-list download/upload center. No longer was there the thrill of the sysop sitting in front of his/her computer all day long and getting excited whenever someone logged in, watching the user's every move, even chatting in real time. All that was gone.
With the excitement gone, many sysops lost interest. Why pay for a separate phone line and dedicate a computer to being nothing more than a free ISP of sorts?
And that is why traditional BBS's have all but vanished from the face of the Earth.