To correct some erroneous or misleading info in previous posts:
There is no bus from Epcot to BC. If you take a bus to Epcot, the only way to get to BC is to walk thru the park, which is not possible before the park opens, and requires admission when it is open.
Bus to DHS is reliable; it just takes longer. Buses start running around 6:30; boats from DHS to BC start around 8 a.m., or 7:30 on EMH mornings. If you take a bus to DHS before the boats have started, a bus will be available to BC; just let the driver on the bus from Pop know where you are going.
All that said, a cab is still the quickest & most convenient, and the 2 are quite close together, so the fare will likely be under $10.