You can fix a leaky gutter joint by caulking it with roofing sealant or by replacing the gutter joint entirely. If you choose to caulk it, make sure the roofing sealant is compatible with the material your gutters are made of (metal, vinyl, etc.). Also, be sure to clean the area around the gutter joint before caulking it to remove all dirt, mildew, and moss.

You could try to fix it by yourself, but it's probably best to call a professional. They will be able to tell if the problem is with the gutter or the roof, and they will be able to fix it properly.