Locate the source of the dripping problem on your gutters. If you haven't already pinpointed the general area the dripping is occurring, pour a bucket of water in the gutters slowly and have a helper or two watch for any water dripping along the gutters.
Examine the section of the gutter that is dripping for any holes or signs of wear.
Clean the holes with a wire brush and sandpaper to get rid of any loose particles. Wash the particles away with water and give them time to dry.
For small holes, mix up a batch of roofing cement, following the directions closely. Apply the roofing cement to the hole and 4 inches around it in each direction. After the cement dries your gutter will no longer leak in that area.
For large holes, apply roofing cement around the hole 4 inches in every direction and cut a piece of flashing that covers all of the cement. Press the flashing down into the cement and put something heavy on top of it to hold it in place while it dries.
Locate the section of the downspout that water is leaking out of.
Disconnect the downspout from the leaking area down. Scrape the old adhesive off with a putty knife. Clean the downspout joint with steel wool and sandpaper to remove all the remaining particles.
Apply a thick bead of silicone caulk around the downspout joint and put the downspout back together using a new connector.