What type of caulk should you use?
Traditional water-based acrylics require both temps and surfaces that are above freezing for at least 3-4 days, so stay away from them in freezing temps unless you care to tent and heat the wall you’re caulking. Instead, look for clear co-polymer rubber or reactive MS Polymer sealants that can be used in freezing temps. Two popular brands are Sashco’s Lexel (a co-polymer rubber) and OSI Quad Max (a reactive MS polymer). Come spring, you can stain over them if the look of the clear isn’t to your liking. Stay clear of silicone. While it can be used in cold temps, it’s not very elastic, nor can it be stained over.
What’s the condition of the surface?
Is there loose or flaking paint or stain at the surface or otherwise degraded wood fibers? Take some sandpaper to just the joint to clean it up before you caulk.
Is there frost on the surface? The surface needs to be frost-free, even with the cold weather caulks. Denatured alcohol usually does the trick; or if you’re desperate, take a blow dryer to the area.
Are there other caulks already present that are failing? If it failed cohesively (split down the middle) and isn’t silicone, you can likely just clean it with some soap and water, insert backer rod if it’s not there already, and reapply the new caulk right over top. If it failed adhesively (pulled away from the edge), you’ll need to remove it. Then, insert backer rod and apply more caulk...