Xbox Support suggests a few steps to try when the light on your power supply turns red. Try the steps in order, and try to turn on your console in between steps:
Step 1: Your cables could be loose/not properly plugged in - With the console turned off, unplug the device from the wall. Then unplug everything from the console. Remove all the AV/HDMI cables, remove the power supply brick (and also, unplug the cable from your power supply if it has removable). Basically, if it unplugs... unplug it. Now, quickly look over all the connections and make sure there isn't any obvious damage. Then plug the cables back in, being sure that they are securely plugged in and aren't loose.
Step 2: Try a different room - The problem may be the outlet you're plugging the console into. Make sure there is no disc in the console, and take the console into another room to try a different outlet.
Step 3: Cool down your power supply - If your power supply brick seems hot, then leave the power supply unplugged from both the console and the wall outlet for a while to allow it time to cool down (approx. 30 minutes). Be sure that the power supply is in a well ventilated area, and that the ventilation holes are not covered. After the brick has cooled down, plug it back into the console and then into the wall outlet.
Step 4: Replace the power supply
I hope this has helped you with your issue. Let us know how it turns out. Good luck.