Putty tape is supposed to be compressed and squeezed out of the joint by screws attaching the trim strip, window etc. I am surprised yours is coming out in a 1" strip which seems to indicate that the whole thing is just dropping out and the fender flare must not have been securely attached in the first place. Remove one or both wheels and use a flashlight to inspect the actual area from which the tape came, but be prepared to remove and redo the fender flare.
Here are two helpful links:
Note that there are two types of putty tape and you'll have to choose which one to use.
I have not used this Eternabond product, but that would be another option and it sounds like your need is a suitable application. Note there are several Eternabond products--double check that you're buying the right one if you decide to go this route.
The cheaper putty tape used by Sunline and most RV manufacturers tends to dry out and crack. The excess squeezed out all over the smooth sides of my 2499 also looks messy and is actually a focal point that speeds up the drying and cracking. Sunline used much thicker tape under the doors and neatly trimmed off the excess, but corners and windows were just left ugly. I used a putty tape tool--much better than a plastic putty knife--to trim off the xs on every single seam and then ran a small bead of polyurethane caulk overtop of every piece of putty tape including the wide ones under the doors. Thanks to Steve Collins for this tip. The trailer looks really sharp now with no more black strips on top of white fiberglass and no more black streaks from door and window putty tape. I'm hoping this is a long lasting fix, but will let you know if it isn't.