Select the color of caulk you want to use in the seams. Most hardware stores carry a variety of colors of caulk, so you should have several options. Take a swatch of the wallpaper to the store with you to match the color as best you can.
Poke a hole in the tip of the caulk. The wider the seams between the wallpaper sheets, the larger the hole will need to be so you can fill it. If the seam is very small, only poke a tiny hole. You can always widen the hole to create a larger caulk bead, and go over the seam if the caulk line is too small. Insert the caulk tube into the caulking gun.
Starting at the top of the first seam, steadily depress the caulking gun's trigger and slowly pull the gun down the seam. Do not push the caulk nozzle into the wall, or you will spread the bead. Do not release the trigger until you reach the bottom of the seam, so you don't break the caulk line. Examine the line to make sure it fills the seam. Widen the hole in the nozzle if you have to.
Repeat with each seam until you have filled them all. Let the caulk dry for 24 hours.
Select a plain white caulk and poke a hole in the nozzle with the knife. You don't have to be as careful about the bead size if you are painting over the wallpaper, because you will flatten the caulk line. Insert the caulk into the caulking gun.
Start at the top of the first seam. Depress the gun trigger and pull the caulk gun down the seam, filling the seam as you go. Run the putty knife over the bead to flatten it out and fill the seam completely.
Repeat with each seam. Let the caulk dry for 24 hours before you start painting.