Beef loses its bright red colour when exposed to the air. Supermarkets sell them in special sealed packs to prevent air getting in so it stays bright red and looks more appealing. If you buy it loose at the butchers, it will be mostly brown by the time you cook with it. The only way to tell if it's ok is a sniff test! It should be absolutely fine the day after the BB date, especially when thoroughly cooked.
(note on sniff tests: sealed packs of meat, fish, salad, veg are usually 'gas flushed', ie a special gas is injected into the pack to preserve the life of the product for a bit longer. When you first open the pack, there might be a horrid smell. That is the gas, and it will dissipate after a minute. Dont reject fresh food too prematurely)
This is why I buy all meat, fish and veg from the butcher/grocer, rather than in sealed packs. At least I can see what I am getting, and there's no nasty surprises once I get home!