How many lights are you going to want to install? What tail/running lights does the trailer have already? (LED lights or halogen?) These are a factor to the response as to make sure you do not draw to much power to trough the trailer harness wiring on the truck.
Both trailers have LED lights. The first utility trailer is open and I was going to install 2 amber grille lights. The second is a box trailer that I am going to install 8 led grille outbreaks a traffic advisor.
Ill tell you what i did with my mower trailer years ago...
I built a 5x10 from the ground up. at the time "leds" werent in existence for trailer lights.
I have 2 self contained oval strobes surface mounted in the fenders... they are wired to a switch near the tongue, the switch is powered via the park light circuit... so they will come on only when the vehicle lights are on. I also connected that switch's power wire with spade connectors. the reason for this if there were ever a need to run the strobes without a vehicle attached i could attach a small externally battery on the trailer and hook the power from the switch to it and then attach a ground wire via an alligator clip or something to the trailer....
you could do the same with yours...
now for your enclosed trailer,
I am not familiar with 7 pin connectors but i believe there is a constant on with those?
you would need to trace that wire, attach the power to your led flasher to that wire and then your leds to it...
other alternatives... small AGM battery and a solar panel with charge controller. (found at most "big box stores")
how long are you planning on running these lights?
If your enclosed has on board battery run the warning off that. LEDs dont draw much and you can leave the trailer stand alone and keep warning on.
For the utility just run a fused line from the car battery to the back of the car (switch it on trailer or in the cab if you feel like it). Be sure to use a connector with waterproof cover (similar plug to a battery tender), and voila. And like Jared's hookup you can use a jump pack or spare battery with the same connector to run it stand alone. As a bonus if you use lights that will run 9-32v you can use a pack of AAs or a 9V to run it.
I just ran a second connector (weatherpak or any brand will do) to my trailers ones a flat bed the other is a box trailer its got just a few light heads all you need is to run 2 pins for power to said light heads or if your running a traffic adviser have a controller on the trailer or run the wires (mine has the controller in the trailer) just size the wires appropriately for the total load your installing (you can do the connector for a ta as well to drive off a cab mounted controller just use a halogen rated controller if driving 2 led traffic advisers and match the head count (some brands have universal head count patturns)
Easiest thing to do is to install normally near rear BTT lights. Ground to mounting screw and wire positive into the parking light circuit on the tail light. No switches, no wires, just turn on parking lights to activate. I would normally use headlights at all times pulling a trailer anyway for added visibility.