Should I fight a ticket I got for riding my bicycle in a crosswalk?
According to Santa Barbara City Ordinances: 

10.52.130 No Bicycles on Sidewalk.
No person shall ride a bicycle on any sidewalk except at a driveway; provided, that a person may ride a bicycle on any area designated by the City Council as a bikeway.  

So, unless this was a designated "bikeway," you were clearly in violation of at least one law. I'm a little unclear on your story, but it also sounds like you crossed against the signal. That would, of course, be a second infraction. 

You mention that you live on a one-way street but were crossing at a four-way intersection. This doesn't make sense. A two-way street crossing a one-way street is a three-way intersection. But I'll do my best to make sense of it. 

If you were riding on the sidewalk on the left side of a two-way street, or if you were riding the sidewalk against traffic on a one-way street; the ticket for riding against traffic also sort of makes sense. It may or may not be entirely correct, but a case could be made.

In short, you were clearly in violation of some law, probably several. If you try to fight it, you may get out of the ticket you were specifically given but you may also end up with several others. 

I would also encourage you to read all of section 10.52 in the attached city ordinances as well as the California Vehicle Code Sections 21200-21212, all of which pertain to bicycling in California. For the best understanding, read the state laws first and the local laws second and look at the local laws as an addendum to the state laws.