Blocking broadcasts from banned viewers
If someone is banned from a streamer's chat, they should also see a notice on the stream window informing them that they're not permitted to see the broadcast along with the active "banned from chat" notification they see in that streamer's chat.
It would make it easier to see for unwanted viewers to be informed that they're currently unwelcomed to that streamer's community.
In addition for the mod/streamer side of the channel, that person's username is completely hidden from the user list of that channel's chat for everyone except the mods and streamer of the channel or that person's username can be separated into its own category called "banned users" but that category would only be visible to the moderators/streamer of the channel as it wouldn't be seen by anyone who isn't a moderator of the channel or the broadcaster of the channel.

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Aussie_iTV commented
When a streamer or mod band someone from the stream, make it so that person cannot access any part of the stream.
Currently they can still watch the stream and access panels. But if the streamer/mods have banned them, then its unlikely they want that interacted with. Or give streamers the ability to chose if their bans remove everything or just chat. -
Fancy_That commented
So we do need some security measure like this.. How about banning the IP?
It isn't fun being stalked online, and have players follow you in the game griefing you from lobby to lobby non-stop for like two days in a row. I already "reported" the individual for this, but he/she will be able to view my stream if they (Twitch) do not take the situation seriously, and he/she will continue to stalk I, and another streamer, in game. It's a pestilence, and harassment. If they're not going to give some sort of measure in somehow banning an individual from accessing another's stream to stalk people, then I hope Twitch perma bans this individual. It's just flat out creepy.
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Anonymous commented
Yes, they could. Which is what they do now, except they're able to completely watch the stream whether they take the time to create a new account/log out or not.
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Anonymous commented
Instead of just having them removed from chat, isn't it sensible and possible to have banned and/or blocked users completely removed from the channel altogether instead of just from chat? This would be separate from time outs and purges, and would be intended to keep harassing viewers from being able to use donate links (unless saved elsewhere) and from watching the stream at all. (Unless a new account is used obviously). Like, maybe a button that allows the streamer/editor of a channel to full-ban or full-block the specified user.