Leak advice needed

I would clear out as much of that old caulking shown in the picture and run a fresh bead of Dicor No-Sag. I'd also look into getting some gutter extensions to prevent the rain from running down the sides of the trailer. I had a small puddle come out from behind the paneling and tracked it down to a tiny pin-hole sized opening in the caulk above the gutters. There was a small air bubble in the caulk when it was applied and after it dried and stretched just a tiny bit it opened up enough to let water get behind the siding, down the wall and out at the base of the door.