What type of climate would I have to have to face a perpetually foggy forest at the base of a mountain range?
There should be almost constant rain, at some level. It may reduce to a drizzle more often than not. Sunshine will kill your fog in a matter of hours, so avoid it.
The mountain range circles the valley trapping cool air inside a "bowl." Or, if it's ouside the range, hills can form some walls of the bowl but you need to prevent winds.
Most of your fog will be generated at night, and you can let the rains stop for several hours at that time. Make sure the air is very humid - not far from and downwind of a large body of water. Infrared cooling from nearby areas which were sunlit during the day will generate "radiation fog" at night, which will roll down into your bowl. Think bordering hills and meadows which are on higher ground than your forest..
Generally keeping the ground warm and wet (above the dewpoint), and the air cool (below the dewpoint) will make a fog. Geothermal heat can warm up your ground, like in Yellowstone Park.
If you let sunlight in, come up with another way to cool the air (snow, thermal inversion, etc.)
I am imagining a world which does not spin or spins very slowly. I believe that climate patterns in such a world would be much more persistent as it is the changing of temperature which dictates the weather. The problem with perpetual fog is sunlight. So as long as the sunlight remains constant there is a chance that the fog in an area generating fog, such as a forest at the base of seaside mountain, could persist.
Disclaimer: I am not a climate scientist. I've just read a few things.
Fog comes from high humidity. A fog bank is, when you boil it down, a cloud that forms at ground level, instead of in the sky.
Mountain ranges tend to promote rain because as the wind forces air over the mountains, the air rises. The resulting cooling temperature and drop in pressure means the air can hold less moisture, so the water precipitates out.
So you need a weather pattern that consistently provides high humidity air, which is then forced (say by a jetstream or similar pattern) over a mountain range. The mountains on the west coast of the US are an example: wind blows eastward, over the ocean, so it absorbs a lot of water. Then it hits the mountain range, and a lot of that moisture falls as rain.