Yes, but it's a bit fiddly.
If, when you are booking, you are either logged in with a BAEC Silver or Gold membership active (BAEC silver and gold get seat selection for free at the time of booking), or you are doing a (not logged in) search for a fully flexible ticket (full fare tickets get free seat selection at the time of booking), then almost at the end of the booking process before entering your credit card details, it lets you pick a seat:
At that point, bring up the seat selection tool, and see how many seats are free on that flight. If not enough, go back and pick a different flight. If so, go back and pick the fare class you really wanted to book on. Expect to take a minute or two per flight you check, by the time you get to that bit of the booking!
Remember that this will only show if you're doing a search when logged in as a BAEC Silver or Gold (other BAEC tiers or other OneWorld status won't work during the booking process), or when doing a fully-flexible search when not logged in.
Otherwise, some travel agents can look it up, and they can also look up how busy the flight is in a general sense (how many tickets for sale in each cabin in each fare class). If you care, it might be easier to get a good travel agent, ring them up, have them look at the loadings then book you onto the flight with the loading / seats you're after.