[Traylaw]

The video provided will absolutely help. By any chance do you use automatic car washes without removing the antenna mast? This is a major cause for damaging the antenna or breaking the seal between the antenna base and the roof.

 

[Traylaw]

Your manual warns you to remove the antenna flexible rod and the car wash usually has a sign asking you to remove it.
As for fixing it, first check to make sure the plastic base is not cracked, give it a little push back and forth to make sure it is ok. You can apply a silicone sealant to the base. The very best way is to remove the base and apply the sealant around the sealing ribs that are around the bolt and reattach the antenna. You can apply your a sealant around the perimeter of the base, but do not cover the drain hole in the back... Of course applying a sealant around the perimeter usually doesn't look very good. If you remove the antenna, take a look around the hole in the roof. Many time the roof gets slightly bent in this area which is why there is a leak.
As for a sealant I prefer a silicone sealant, note however there are many kinds of silicone sealants with different properties. The one below in the photo is excellent for this job. It won't harm metal, extremely flexible and provides a very strong bond and design to be constantly in water.

 

[Traylaw]

You don't actually need to remove the headliner, just carefully pull down the rear of the liner to gain access to the nut holding the antenna.
Sticking a shark fin cover probably would be the easiest. I think it's a matter of what you're comfortable doing. Of course the dealer can also fix it.....it just going to cost money.

 

[Traylaw]

Go from the mildest and work your way up. Try on a small spot first. Hydrogen peroxide, Oxyclean, Clorox. Google bleaching out water stains for instructions.