Seeing this late, let me give an answer which everybody now is surely aware of. The answer is yes for every positive a,b >0. The reason is that log Beta (a,b) is infinitely divisible additively so that B(a,b) is infinitely divisible multiplicatively. To see the ID property for the log, compute the Mellin transform with the Gamma function (as suggested above) and apply the Malmsten formula for the latter, which is in this case a Lévy-Khintchine formula.

Seeing this late, let me give an answer which everybody now is surely aware of. The answer is yes for every positive $a,b >0$. The reason is that $\log\operatorname{Beta}(a,b)$ is infinitely divisible additively so that $B(a,b)$ is infinitely divisible multiplicatively. To see the ID property for the $\log$, compute the Mellin transform with the Gamma function (as suggested above) and apply the Malmsten formula for the latter, which is in this case a Lévy-Khintchine formula.