The B - Cl bond is more polar. And B is the partial positive, and Cl is the partial negative atom.
Work Step by Step
According to the picture 6.8 (Page 263), the difference between the electronegativity of the boron and the chlorine (2.7 - 1.9 = 0.8) is greater than the same for boron and carbon (2.4 - 1.9 = 0.5). Since that difference is proportional to the polarity of the bond, the B-Cl will be the more polar one. The less electronegative one is the boron (1.9), so it will be the partial positive atom, and the chlorine will be the partial negative.