Weigh your trailer one time you are headed out on a trip to make sure you aren't exceeding the safe weight limits of your tow vehicle. Be sure the tires and brakes on your tow vehicle are kept in good working order since towing a trailer places higher than normal strain on them. Poke your head under the hitch receiver on your tow vehicle periodically to make sure everything is secure. Sometimes mounting bolts work loose or frame welds crack. If the hitch comes off your tow vehicle your trailer is probably a goner.
Propane is a potential safety concern.
Don't travel with the bottles open. If you ever smell propane you should close the bottles and investigate for the cause of a leak. On a hot day the tanks can vent extra pressure which will make you smell propane, but ordinarily the system is completely closed and you should not smell anything.