What’s the fastest way to earn Prisoner’s Honey

The fastest way I've found so far is in Mahogany Hall; it requires a very high Persuasive skill. If you train the Louche Devil and teach him Philosophy, you'll get 111 Nevercold Brass Slivers per action and increase your Pygmalion quality. When Pygmalion reaches 17 or 18, you should be safe to take him out to a Society dinner, which gives you 1000 bonus slivers. ... You can then re-train him as often as you like to rack up lots of Echoes.

If you want to do it with your Dangerous, head to the third coil of the Labyrinth of Tigers. "Avoid becoming an exhibit yourself" and "Fight back" for ~1 Echo per action. (Most of the actions in the third coil earn ~1 Echo per action, but they require a very high Persuasive as well.)

Generally there are quests around the Flit or Wolfstack Docks that will do some good, and obviously playing opportunity cards that will commit dodgy deeds or frustrate the Constables are likely to help.

As long as you have a Connected: Criminals of at least three, then you can go to Mrs Plenty's Carnival. Buy or steal some carnival tickets, then go to the sideshows. From there you should be able to exchange tickets for Connected: Criminals by associating with the criminals. This should enable you to get your Connected up to 10 and use the key.

Also, here is a large list of things that will increase your Connected: Criminals. http://fallenlondon.wikia.com/wiki/Category:Criminals

Good luck.