I also contacted BART via the customer service email listed on the BART website and received this answer while out for the holidays:
Bike trailers are permitted on BART, with the same basic restrictions as bicycles (no bikes in the first car, or on trains passing through downtown San Francisco, Oakland or Berkeley during rush hours). The only additonal restriction would simply be extra awareness, as you will be using more space then a regular bike, and to avoid crowded trains, even if it is not during the restricted hours. Thanks for choosing BART.
BART (Bay Area Rapid Transit)
Transit Information Center
Tel. 510-465-2278 (live operators)
6AM to 11PM - 7/Days"
I called BART and the person I spoke with was very nice and quite informative.
So, first thing is that you need to follow the standard BART bicycle rules. Basically, there are a number of bicycle restrictions regarding peak commuting hours. You're pretty much committed to following those rules.
According to the rep I spoke with, during "non-peak" hours, you may or may not be allowed to use a trailer. The station masters will look at the length of your set-up. So, during low demand periods, you're probably ok if your trailer and bike are detached and compact as possible.
So the message I got is this. You will not be able to commute during peak hours with a trailer. You're probably ok with some trailers during non-peak times; except at times when a big event is occurring in the city.
I'd suggest that you pick an off-peak time when it's not terribly inconvenient and try out your bike trailer set-up. Also, according to the BART rep, if you're organized about things and your trailer is basically a large piece of luggage, you're probably good to go during the off-peak hours.