I assume that this is a 120V system (you live in North America) and that it's probably a 15A circuit breaker. That is the normal maximum for a 120V line. A 15A breaker trips at an instantaneous load of 15A, but it's only rated for 12A continuous which is what you have in your case. You can't limit the amount of current going to the power supply because it draws what it needs to power your system. If you could put a limit on it, instead of the breaker tripping, the computer would shut off which would give you the same result minus having to run to the breaker box to reset the breaker.
However you may have a faulty 15A breaker which is tripping before it hits 12A/15A. An electrician can replace the breaker. Actually you could probably do it, but there are issues with insurance and you working on your electrical system and not being an electrician.
If however, it is actually hitting the 15A on that circuit, your only other choice is to plug the computer into one circuit and move everything else to another circuit. Depending on how your house is wired, you could have more than a single set of outlets on that circuit. You could even have the lights in that room on the same circuit. To find out if you have another plug in that room that isn't on the same circuit, turn the breaker off that keeps tripping. Using something like a lamp (if you don't have a multimeter) plug it into different power outlets in the room until you find one that lights the lamp. This outlet is on a different circuit. Then plug all the other stuff into that outlet. Hopefully it's relatively close to your computer or you'll have to find a way to hide ugly extension cords.