I am a fellow applicant and know Hopkins well, BI reasonably well, and know nothing about Yale. What I know is from first hand experience and what I hear most frequently about the programs.
Repute: Hopkins > BIDMC > Yale
In this list, Hopkins is the sure winner on academic repute and name branding. BIDMC comes a far-ish second on the basis of it being the 'third' Harvard anesthesia program. Yale school of medicine or the program just doesn't carry the same amount of clout that the Harvard/Hopkins name does. For some reason Yale's residency programs are great but not considered in the same academic league as say, Columbia U or UCSF.
Case complexity: Hopkins>BI ?Yale
Hopkins gets all the $hit that has hit the fan in one of the nation's worst neighbourhoods (East Baltimore). Moreover, because of the name, the surgeons get referrals from all over the nation and internationally, so you get to see some very sick/complicated patients. At Hopkins you really are the king of the jungle for all kinds of complicated cases in the DMV area (the other major competitor, U Maryland is a great hospital but the residency progarm is far from great). Folks from BI hate to be called Harvard-lite, but the general concensus is that they live in the shadow of the two academic behemoths, the General and the Brigham and cases get divvied up with such saturation of brand-name institutions (Tufts/NEMC takes its share as well). I don't know much about Yale but being the top dog in the CT region, I am guessing that they'd get some fascinating stuff as well.
Teaching: Hopkins = BI ?Yale
Hopkins has its college days thing going on where you get a day off on Thur to study or attend whatever they want to teach you. The word in Boston is that BI is the least stressful of the three Harvard programs and teaching is good. They present a 'friendly' image of their program where the anesthesiologists and surgeons are buddies etc etc.
Research: Neither Hopkins nor BI has a research oriented Chair or PD (the BI chair is from Hopkins). These institutions do a lot of basic sciences research at faculty level and I expect to get attacked for saying this, but these are not research friendly places at a resident level. That said, I am speaking from my perspective, that of a well-established researcher, and if you are at the stage of 'maybe I am interested in doing some research, I worked in a lab for 2 months between undergrad and med school, so lets find out if someone is doing something with receptors or channels (i.e. flexibility and wanting to explore options), then you will find loads of opportunities at all of these places.
My $2s.