The walls around your bay window are damp why?

If the wetness is just below the window, it's probably condensation.
Is the glass fogged up most of the time?
The easiest fix for this is to run a dehumidifier in the vicinity of the window
The other two options are adding storm Windows or replacing the glass with thermal panes.

It could be that the connection between the window and the house is leaky.
This would cause the walls on the sides of the window to be colder then the air in the house and condensation will form on the walls.
Do the walls feel cold next to the window?
This is the easy one, just calk around the window.

The last option is that the roof over the bay window is leaking.
I'm assuming it has a little apron roof over it that sticks out of the exterior wall.
but weather it does or not, it's probably the flashing at the upper joint between the house and the window.
You can't calk that joint. It needs to be open so that any moisture that gets under the siding can get out.
You can look at it yourself, check for cracks or openings above the window and especially at the corners.
It may be that the wind is blowing water up this joint and over the flashing.
Or if snow and ice are building up on top of the window, the melt may be working it's way in.
The joint itself should be tight and you should see a little strip of metal flashing (copper lead or aluminum) sticking out.
You might want to get a pro to look at this.