I would seriously doubt that the monitors are a significant source of radiation. First of all, it's well established that LCDs emit far less than the old CRTs that we all grew up with. The early Mac and PC screens emitted rather high levels compared to the CRTs that came out after new standards in the 1990s, and LCDs are even below those. The dermatologist might have carried over experience from CRT levels of radiation that do not apply to today's LCD monitors.
Also, the light isn't even pointing directly at you like a CRT. The backlight is at the end of the display, and the light is directed across the screen, not out at you. A large diffuser evens out the light, and it filters out the front. Obviously, by that time, the light's been reflected and diffused quite a bit.
OK, so then where might this UV have come from? I have a guess. What is your workspace like? Are there large banks of fluorescent bulbs above you in your office? If so, I think there is a higher chance that you may be receiving a UV dosage from those lights. It's much more direct, and at a much higher level than any monitor. I have seen photo prints fade under normal fluorescent lighting as if they were exposed to direct sunlight. If there are one or more defective bulbs, maybe that could cause a problem.
Even more likely: If you happen to work in a facility where there are high-intensity metal halide bulbs, and any are broken or defective, it is possible to suffer serious UV-related effects.
Read this article as an example.
"In February, a broken bulb was discovered in a middle school gymnasium in Haywood County, Tenn., after 40 children complained of burning eyes and skin rashes, said Timothy Jones, a state epidemiologist."