Not really a solution, but if you turn it off and back on over and over (like 30-50 times) there's a good chance it might come on and stay on eventually. I have two ViewSonics, different models, that both do that. I've been using that work-around for over a year on both of them.
The real answers are: take/send the monitor in for repair, that'll probably start at around $100, or get a new backlight/inverter and install it yourself, or get a new monitor.
If you are real ambitious, you can open up the monitor and look at the inverter board; if the capacitors are bulging on top then it's the inverter. If the capacitors are not bulging on top it still might be the inverter. I was not able to find an inverter for your monitor anywhere but the part number is 52-5W273003G000. I did find the main board online for about $20; its part number is 70-V7030102G500.
Also found the service manual
which is where I got the part numbers. There is an exploded diagram with accompanying parts list. The diagram labels each part with a circle with two numbers in it. The top number is supposed to be the item number as listed in the table. The bottom number is how many of that particular part there is. But beware; the "exploded parts list" is numbered in reverse compared to the exploded diagram. So what is labeled part 1 in the diagram is really part 24 on the list, part 2 is part 23, etc.
Many people have had success replacing the capacitors on their inverter boards for just a few bucks; check out this thread
(also look at the one linked in the second postfrom badcaps.net. If you are handy with a soldering iron it's actually pretty easy.