(I know, but it happens)Analyze their credit report and if you feel comfortable, do a background check and make sure their criminal history checks out.Another key factor is employment history.

But, the landlord could be on the hook for a tenant's actions, most notably serious criminal offenses.

STAY AWAY FROM THAT CRIMINAL! 

Pretty much the more you know, the more scary cyber security becomes.  

I had a warrant issued and am being charged with "criminal housing violation". 

Friend got it with this at a vacation home.Forget the billing and fix the problem:First take better pictures of the meter and the connection for evidence.Get an electrical contractor to come out, disconnect the illegal connection and install a security box with a lock so there's no access to the power  connects.Write a letter, send it Certified to the neighbor with copies of the tampering that this is criminal fraud and theft and if it occurs again, they will be prosecuted - - clearly note a copy has been sent to the Utility Company.You might research the owner of the property and send a copy to them also (in case it might be rented).You didn't start this but need to control it.

They offer little protection if someone is injured due to negligence or criminal actions.

He gets deported through ICE as a criminal alien.

I do not see anything mentioned that would convince me not to do a credit and criminal check as the wife would need to be added to lease if approved.  

-Check their criminal background (some credit reports will show this, but others don't)-Verify their income (run the numbers in terms of their income, the rent you charge, their other current bills, etc.)