As for some actually helpful information, because the bans are so random, it's hard to know anything for sure. Even if you go back online and find that you're not banned, you still could be on a "pending ban" list and may get banned later.
If current theories are correct, Nintendo is banning based on Title IDs being played on your 3DS, and doesn't have a surefire way of knowing whether or not you're running CFW.
If you're running a homebrew like FBI off of the home menu, from an installed CIA, that Title ID shows up in your Activity Log and on your Friend's List. Nintendo may be banning based on this.
So your safest bet, in my opinion, if you want to play online is:
1) Disable sharing data with Nintendo in your Spotpass settings. Also go into your Friend List settings and disable sharing which game you've been playing.
2) If you're on 11.4, set up an exploit like oothax so you can launch the homebrew launcher.
11.3 and below can use soundhax instead.
3) Delete any homebrew titles from your 3DS that have been installed from CIAs. You can use FBI for this, use the 3dsx version. You could also delete them from your system settings, but some people say this still leaves traces of your previously installed title ids behind. You're better off uninstalling them with FBI.
4) Delete any CIA titles you have installed that you haven't gotten legitimately.
We don't know whether or not this is actually necessary yet. But there's a possibility that Nintendo could see whether you are using a cart or CIA. They could be checking title ids against a list of what you should own from the eShop. They could also potentially see if you're playing a game that was only released on a cart, as a CIA. They could also be checking whether the title you're playing matches your console's region. The fact is there could be a large number of things they are checking for and nobody in the homebrew community has a list of what they are. So if you want to live life dangerously, leave them installed. If you want to play it safe, delete them.
5) Wipe your activity log (there's a 3DS homebrew app called Cthulu that can do this. Use the 3dsx version.)
6) You THEORETICALLY should be safe to play online (BUT we don't have any solid proof of this, and if you get banned later we have no way of knowing whether it was because you went back online or if you were already on a list to be banned...)