Usually broadband just means "not dialup".
If you have even the lowest end DSL, it'll probably be playable, and anything faster than a megabit should be more than sufficient.
I played the open beta on a 1mb/s connection without any problems.
And wireless just tends to be buggy, especially with stuff like this where a few lost packets can make you jump around the screen and lag a lot.
Putting in a caveat about wireless is just a way for them to wash their hands of your local network problems.
gaming.stackexchange.com/users/479/Satanicpuppy
What qualifies as "Broadband"?
The term broadband refers to a telecommunications signal or device of greater bandwidth, in some sense, than another standard or usual signal or device (and the broader the band, the greater the capacity for traffic).
Source
Any good connection to the internet.
Diablo 3, different from diablo 2 and diablo.
can ONLY be played with internet connection (without being dial-up) Does DSL-3Mb connection count?
Digital subscriber line (DSL, originally digital subscriber loop) is a family of technologies that provide internet access by transmitting digital data over the wires of a local telephone network.
The 1980s saw the development of techniques for broadband communications that allowed the limit to be greatly extended.
Source
And what could possibly be the issue with Wireless?
The only issue about Wireless connection is the interference by walls and other stuff between your device and your antenna.
gaming.stackexchange.com/users/12936/Michel
I would think your DSL-3Mb would suffice.
With regards to Wireless, it will probably be playable, but it can be very "unstable" because of latency, interference and signal strength issues.
And 802.11b has a theoretical max of 11 Mbit/s, which can be lowered because of interference and signal strength.
I would suspect Blizzard added that caveat to be able to say during support, that people should try a wired connection and see if that helps.
gaming.stackexchange.com/users/8017/Holger
By Broadband, they simply mean 'faster than dialup'.
The 'some wireless connections are not true Broadband' is not referring to Wi-Fi networking, but rather to wireless internet connections provided by cellular providers.
Many of these are still available at speeds well below 3G - much closer in fact to the sort of speeds you'd get over old Dialup, and that's before taking questions of reliability into account.
The warning is primarily aimed at users who expect a high quality gaming experience over this sort of connection.
Not Wi-Fi.
gaming.stackexchange.com/users/3129/LessPop_MoreFizz
And what could possibly be the issue with Wireless?
If you are talking about 3G/4G then the answer is that its simply not fast enough.
If we are talking about 802.11N then that of course is supported.
3G/4G is fine when you are talking small TCP packets that happen when you refresh a page; Diablo III will require continuous packets.
gaming.stackexchange.com/users/24871/Ramhound
The least connection I have used thus far is a personal hotspot from my 3G iPhone providing a wireless signal for my laptop.
The game was quite playable but I wouldn't try hardcore that way :-)
gaming.stackexchange.com/users/10260/Tharius
According to  Diablo 3 System Requirements , we need a broadband connection.    What qualifies as "Broadband"?    Does DSL-3Mb connection count?    What could possibly be the issue with Wireless?
What does Blizzard mean by broadband?
gaming.stackexchange.com
<diablo-3><system-requirements>
gaming.stackexchange.com/questions/65394
gaming.stackexchange.com/users/17761/Mike
Broadband means not dial up. It refers to a device of greater bandwidth than the standard signal. In simple words, broandband means not dialup. Basically any good connection to the internet.
A wider definition of what broadband means, is given.