While it has not yet been proven that walnuts can actually lower blood pressure, they have been known to lower cholesterol levels in those who consume these nuts on a regular basis. It is highly unlikely that walnuts could cause high blood pressure though, especially considering all of the nutrients that they have to offer.
Walnuts also have a very low number of carbohydrates and a few different “healthy†types of fat which are important for the body. If anything those who consume a significant amount of walnuts on a regular basis might actually find that their blood pressure level decreases over time.