Vertical stabilizers with anhedral improves directional stability - you could also say that it counters dutch roll tendency.
The physics behind this is in fact quite simple, it is purely geometrical.
If we assume an A/C heading a little left of its track (sideslips or yaws a little in a gust or so), then on a swept back wing the right wing will present a greater span to the airflow than the left wing. It means more lift on the right wing than the left one. That will initiate a roll to the left which is undesireable since we already had an unwanted yaw to the left.
The horizontal stabilizer with anhedral, however, it will increase its negative angle of attack on its right side, and decrease angle of attack on the left side. That creates a roll force which is opposite the unwanted roll force from the swept back wing. Correctly designed it can counter it 100%.
Then why do so many airliners have a horizontal stabilizer with dihedrad? The answer is structural weight. A few hundred pounds extra weight would mean a few less paying pax, everything else equal.
Take a B737. It saves at lot of weight having the horizontal stab connected directly to the fuselage instead of having it connected to the fin. But the horizontal stab must be out of the engine exhaust, therefore dihedral is definitely needed. It cannot be straight either since it would moke it far too prone to tubolence from the wing, especially with flaps extended.
So the "fix" to solve the dutch roll problem on planes like the B737 is the automatic yaw damper, a gyro operated computer which counters the yaw instantly by actuating the rudder.
Planes with T-tail have already "paid" the weight penalty. So on T-tail planes we most often see horizontals stabs with anhedral.
Back in the 60'es the Germans created a biz jet, the Hansa Jet, with a swept forward wing (it never was successful). One of the reason for that was that dutch roll tendensy was actually killed at the root, as opposed to swept back wings (another reason was to have the cabin totally forward and unobstructed by the wing main spar).
Forward sweep has, however, a lot of structural disadvantages which can only be countered by accepting weight penalties. In addition, on large airliners forward sweep would mean that the wing main spar and the main landing gear would compete for the same room, meaning that the landing gear most likely would have to be podded on the wing like on a Tu-134 = increased drag.
It's all compromises, compromises and even more compromises.
Regards, Preben Norholm
Always keep your number of landings equal to your number of take-offs