Do you have your power supply directly connected into a wall outlet instead of a power strip? I ask because this can sometimes cause the power supply to not function properly.
Usually the power supply turning red continuously means that it's just gone bad in some way and if that's the case you can always get it exchanged (if you have a warranty) or replaced if you don't have a warranty (it costs $35 USD and can be purchased here). Before purchasing a new one or even exchanging it out, you may want to to consider trying a friends power supply or taking it to a local game store that sells XBOX 360's and ask them if they could test one of their power supplies with your console to see if it works. Those are really the only surefire ways to know if there's something wrong with the power supply or if it's a larger issue with the console.