B.A.B.E is located at 

Asylum CoLab 
Faneuil Hall, 200 State Street, Suite C-1   
Boston, MA 02109

The CoLab is located at the Rose Kennedy Greenway end of Faneuil Hall, on the second floor of the modern two-story building between the end of the istoric marketplace and the Greenway. If you stand with your back to Ned Devine's and look towards the Greenway, you'll see it.

Parking

There is validating parking in the Haymarket parking garage.  It’s $3 for 3 hours.  Validate at the front desk at the CoLab.  The entrance to the garage is on Sudbury St, around the corner from Union St.  The front of the garage looks like this.