What's the fastest way to earn Connected: Criminals?

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.