This is a bit difficult to fix if you are not knowledgeable with auto wiring but here goes!
Download the 2003 Jeep KJ Service Manual below.
I presume this is a Gasser with Manual Transmission.
Go to Page 8W-21-5.....you will see the Back-up Lamp Switch at the bottom of the page.
+12 volts goes onto the switch from fuse #39 via a Dark Blue Wire with a White Stripe....this should all be OK!
To check if the Switch itself is bad, pull off the Brown/Light blue wire from pin #2 of the switch so the switch should be floating in mid air. If switching ignition On and operating the switch blows the fuse then the switch itself has a fault inside it!
Then the +12 volts leaves the Back-Up Lamp Switch via a Brown Wire with a Light Green Stripe......somewhere here is a short to ground!
This Brown/Light Green wire goes through a lot of connectors before finding its way to the Left and Right Tail/stop lamps and Trailer Tow Connector.
You will need a Digital Multimeter....a cheap one will do as major accuracy is not needed. With the meter set to Ohms and ignition
OFF, place one meter lead onto the Brown/Light Green wire at Back-Up Lamp switch pin 2 and the other lead firmly grounded to chassis/engine. Make sure you do not touch the lead ends with bare hands as your internal resistance with affect the reading.
You will expect to get some reading of resistance..a few Kilo Ohms, as you have the Back up lamps fitted. Remove these lamps and you should not have any reading but probably will have a reading of only a few Ohms as there is a short somewhere!
Find the locations of all the connectors that this Brown/Light Green wire goes through...the connector locations are shown near the end of the Wiring diagrams.
Start at the top ie. disconnect connector C310 while keeping your meter lead on pin #2 of the Back up Lamp switch. If the meter reading jumps up from the low-ohms being shown to a high ohms reading then the short is above C310.
If not, continue down this line disconnecting the connectors C306, C207 etc. until the shorted section is found.
So let's say that if C201 is opened up and you get a good reading above C201 but a bad reading below C201 then the wire section between C201 to C100 may be shorting to ground or the section between C100 to C105 is shorting etc.
Once you find the shorting length of wire, try physically trace the wire to look for shorting to engine/chassis. If the wire's insulation is damaged...try insulate it. If you cannot find where the wire is damaged because it disappeared into some hard to reach spot then you need to cut off the bad wiring at both ends and replace it with a new section by soldering and insulating it and wrap it up neatly with the existing harness.
If you do not wish to buy a meter, try get a DC 10 Amp Circuit Breaker to fit into Fuse#39's socket. Then you can disconnect the connectors one by one until the circuit breaker stops tripping.
I have found about 15 shorting wires on my Jeep after a front-end collision...most of the bad wires were unreachable so I had to cut off the wiring at all ends and replace with new wire!
Have fun!
www.colorado4wheel.com/manuals/Jeep/KJ/