This is the best time for starting! You'll get steady work for at least two years, and by then the construction situation should have improved enough to keep the apprentices busy. By the time you top out, work will be going stronger and you will have more job security.

The down side of this economic slump for your apprenticeship is that you will be pushed for production, and lose out on having co-workers spend extra time on the job teaching you things. Most everyone will be sort of under the gun for a while, and the journeymen you work with will be 'survivors'...so most likely looking at short term gains, foregoing the time and expense of doing a professional job in order to keep their jobs. Also, the contractors will be using you to do the more repetitious and easily trained tasks, so skill acquisition will be a bit constrained...at least until the competition for jobs loosens up.
If your locale keeps apprentices working in one shop for several years, you might be stuck doing the same old crap, and whatever you do best will be something they stick you with. So, keep in mind that you might eventually have to un-learn bad habits picked up on the job, and lobby hard for new job experiences if they try to typecast you.

But...first things first...you still have to actually get into the program!

High school diplomas are a 'minimum' requirement, college records will suffice. But please don't think that a college degree will be a passport into the program. They will be looking for some proof of your having the right kind of attitude, and a history of a good work ethic, with emphasis on mechanical skills. They will be looking for some verification that you can actually deal with a physically demanding job and get along with all sorts of people under adverse conditions.

Best of luck, Macwire!