Since it's in group 13, B has 3 valence electrons, while F (in group 17, a Halogen) has 7.
Reactivity can be based on how easily an atom can lose or gain electrons to form a complete, valence shell, with 2 or 8 electrons in it. Atoms always want a full valence shell: this makes them stable.
Since F has 7 electrons, it only needs to gain one electron to form a full valence shell of 8 electrons, making it highly reactive.
B has only 3 valence electrons. This means that (usually) it would have to lose 3 electrons to form a valence shell with 2 electrons, or it would have to gain 5 electrons to form a valence shell with 8 electrons.
Because of this, B is more stable.
If you want to find out more about valence shells and how they connect to reactivity: http://www.bbc.co.uk/schools/gcsebitesize/science/add_ocr_21c/chemical_patterns/patternsrev5.shtml
^Click through it - it has very useful information