What I think puts people off casual browsing of old-school BBSs is that almost all of them require 'user registration' before you can actually log on.
These days, I think 'guest' access is mandatory.
I'm a strange person, I won't use 'modern' gadgets like the PLIP box to get my Amigas online. For me, Retro is about re-enjoying the classic machines along with the era-appropriate peripherals (where possible).
What I've been doing for a while now is using a Null-Modem cable to my main PC and running a custom 'BBS' type app that effectively offers file browsing of a local folder on the PC along with XModem upload/download. This is a good stop-gap to get stuff on/off my Amiga's hard drive. I use JComm on the Amiga as the terminal client.
Lately, I've been playing with AmigaNOS, which is an all-in-one TCP/IP over serial application that runs on the Amiga. Connecting to a Linux machine (again over the null-modem) with slattach running on the Linux box gives me a point-to-point TCP/IP connection. Googling AmigaNOS instructions on how to do this has shown that the knowledge has been lost over time, so I'm documenting it heavily for a future blog post.
The downside of slattach is that it doesn't work with iptables so I can't route the amiga traffic to the outside world. So the next thing is a cheap old Cisco Router with a serial port on it that will route serial-to-network properly. I'm waiting for that to arrive at the moment. End goal on this is multiple serial ports to ethernet for all my retro machines.
I love Kickstart/Workbench 1.3, but bigger TCP/IP stacks like AmiTCP or Miami don't play well (if at all) with it. Hence exploring this esoteric method with AmigaNOS.
OK, this probably isn't the answer you were looking for :) So... for less crazy people, the answer is: A PLIP Box, Kickstart 3.x, and Miami or AmiTCP.