First, I would suggest to find the cause of the water intrusion. If it's from road spray, seal the areas with a sturdy solution like sheet metal, using appropriate sealers to stop further water intrusion. If the water is coming down the interior of the wall frame, seal the roof and any other possible leak areas and then, repair as above.
Trying to tape together deteriorating wood to siding or trim edges is going to drive you nuts. As the trailer is towed on the road, those surfaces are going to move and any tape is going to fail, eventually. If tape is the only solution, get some of the reinforced "gaffer" tape that is sold to repair coroplast foam panels. It is typically sold at RV dealers but you can get it online. A bit more expensive is Gorilla brand tape, which has a high bond strength.