The blood-brain barrer exists to protect the sensitive central nervous system from the outside. We do not always want everything in our blood to be able to reach our brain! In fact, if certain things in our blood were able to flow out into the central nervous system, some significant problems can result.
The most obvious problem with a weakend blood-brain barrier is infection. Encephalitis is an infection of the brain, and it should always be treated as life-threatening. In order for such an infection to take place, the brain must have either been exposed to the outside, or the blood-brain barrier must have been weakened. Considering how many bacteria, viruses, and fungi take hold in the latter state, it is clear how necessary the blood-brain barrier is to maintaining our good health!
A second consequence is almost the opposite situation. The central nervous system is termed "immunoprivileged" due to its generally being sequestered from the immune system. This status does become problematic, though, because immune cells are not "trained" to avoid attacking antigens of the central nervous system. Because these particular self-antigens are not necessarily tolerated by the immune system, their exposure due to a weakened blood-brain barrier can lead to autoimmune disease.
There are other consequences of a weakened blood brain barrier, such as having less control over the potential and equilibrium of the many solutes and ions in the blood.