Whether economic times are good or bad, once a firm does decide to "clean house" your position is only as secure as your last review. If your review has not been particularly favorable, the chances of you staying with the firm are not too promising. Throw in an unstable economy, and my guess is that you will soon be dusting off your resume.
On the other hand, if you are receiving stellar reviews and the partners and/or senior associates are asking for you to work with them on their client matters, I would guess that even with a house cleaning, you are still going to be part of the firm.
The chances of this kind of house cleaning affecting first-year associates is less at the kind of firm you're joining. Major New York firms generally have a fairly accurate estimate of the work they will have for the next few years and know how many first year attorneys they need to have on board.
Even if they have misjudged their need for new attorneys, major law firms generally try to avoid laying off first-year associates for two reasons. First of all, these new attorneys haven't been around long enough for partners and senior associates to judge the quality of their work. Unless the firm is in dire economic straits, or you are absolutely, positively not able to grasp what you are supposed to be doing, it is rare to see such a junior associate being let go. Secondly, letting go of a first-year associate also harms the firm's on-campus recruiting program much more than when it lays off mid-level and senior associates. For those reasons, first-years tend to be exempted from layoffs.
Although one can never be truly 100 percent safe, my guess is that you do have a modicum of job security simply because of the kind of firm you are joining.
The important thing to remember is that as long as you are doing a good job, you probably do not have to worry at this time. However, if you do find that you are having problems with your assignments, or that the assignments are slow to come your way, you need to take some immediate action. First, speak with the partner you report to and try to work out the problems you are having. If it does not seem that things are going to get better, get yourself out on the job market before you become a layoff statistic. Best wishes to you.
Summary: In the wake of the associate layoffs at other firms, I was wondering how safe my position would be if my firm decided to clean house