Sealing edges of trailer. What putty or caulking do you use?

It could be Butyl or putty tape. Putty tape is made from clay. It is perfectly normal for it to ooze out over time. Use a plastic scraper to carefully remove the excess. To do a full repair requires removing the entire piece of trim.

If you do remove the trim you may want to consider a relatively new product; http://www.sealdesignllc.com/catalog/seal-tite-corner-seal/