Space is full of radiation -- that's a given. Even in our part of space, a ship would need to have very good radiation shielding for its crew to survive very long. Just being in the International Space Station for a few months increases a person's cancer risk significantly, and any manned ship to Mars is going to need really good radiation shielding if we want the crew to live long lives once they get back. Clearly, Starfleet vessels must have considerably better radiation shielding than we have today.
Yes, the radiation in the Central Bulge might be somewhat higher, but remember, radiation exposure is cumulative. Any ship with good enough shielding to survive, say, 10 years in space at normal galactic-disk radiation levels would be equally able to survive 1 year in space at levels 10 times greater. With the enhanced warp drive the "Demon" ship was using, it could've cut through the Central Bulge in weeks or months.
Really, the greatest radiation hazards in space don't come from ordinary stars going about their business. Sure, a stellar flare gives off a burst of radiation, but that's really only hazardous if you're in-system, much closer than a light-year. The really serious hazards come from supernovae and distant gamma-ray bursts. But the only stars that go supernova are the really big ones that burn out fast. The Central Bulge has very little star formation occurring in it (except within a few parsecs of the nucleus, perhaps), so just about all its supernova-capable stars have long since died out. And most of the stars would be old and settled down so they wouldn't flare too much either. There may be occasional massive radiation bursts resulting from bursts of star formation (and destruction) near the nucleus, but that means "occasional" on the time frame of hundreds of millions of years, and nothing of the sort seems to be going on lately.
Really, the arms are the dangerous places to be, in the long term. Spiral arms are waves of new star formation, which is why they're bright and blue. They're the places where you get the big, short-lived stars that go supernova. The Bulge is pretty sedate in comparison -- except, again, in that very narrow zone right up close to the central black hole itself. Most of the Bulge is essentially like a really big globular cluster, which is Dullsville in astrophysical terms.
