Edit 2012.05.31 I decided not to wait any longer for Noam Elkies. Here is my idea of a lower bound argument. It can probably be extended to open balls; I prefer to use compactness and closed balls for simplicity.
Let there be a covering of the closed unit ball by finitely many closed balls of radius 1/2. Any covering ball which contains the center, call it c, of the unit ball contains at most one point, call it p, on the boundary B of the unit ball. Since B minus p is open with respect to B, p is contained in one of the other covering balls which does not contain c. So we can assume the boundary B is covered by balls none of which contain c. The covering now has a finite number of balls which cover B plus at least one more ball covering c, and perhaps others.
Now replace the covering above with a new (perhaps identical) covering: shift each ball toward or away from c so as to maximize its intersection with B. This places each covering ball center at distance sqrt(3)/2 from c. B is still covered, and this new covering along with a ball of radius 1/2 placed with its center also at c is another (perhaps the same) covering with the same or fewer number of balls. Thus the posted problem is (essentially) the same as optimally covering B with caps of spherical radius of 30 degrees.
Elsewhere I noted Neil Sloane had a cover of a 3-d sphere with 20 caps each of radius slightly less than 30 degrees. I now claim anupper bound of 21 for the posted problem. Assuming Sloane's expertise with sphere packing, I expect 21 to be an exact bound.You can ask him for the covering number for dimensions greater than 3.END Edit 2012.05.31