Hmmm, well I think one of the problems here is that people think that there is only one secret at Area-51. Given the size of the facility not counting what might be underground, I'd imagine it works on a lot of differant projects.
As for why this would continue despite the nature of the base being "outed" is simply because research facilities and such do not grow on trees, and simply building another one would be really expensive. If you have the facilities, it's stupid not to use them.
The problem with all of the alien stuff is that I still can't think of any sensible reason why this would be kept secret. All of this stuff about "it would cause a public panic" is BS, ditto for some of the stuff about "cold war paranoia" simply because a reveal like aliens would probably have ended the Cold War, one thing that can unify humanity is a common interest or potential threat. I suppose fear of sudden, radical change might have played into it, but I just can't see it to be honest. That's the problem with most conspiricy theories: motivation.
If I had to guess what is going on in Area 51 it would simply be that the people there are all scientists with goverment grants who are happy to have access to the facilities and do their little pet projects. Nobody wants to risk the resources of finance, and they are all doing what they want to do, so there are no real leaks by them or their people. Most of the rank and file soldiers and employees probably don't have access to the labs, which are all working on differant things.
It's possible you might have Aliens in there or something, but honestly I wouldn't think so.
Another reason for secrecy of course is peacenik liberalism. This is a military facility where a lot of research is doubtlessly being done in how to kill massive numbers of people extremely quickly. People already freak out about nuclear weapons and getting rid of them, as well as blockading research for sub-tactical nukes which could be used more freely. In a society where left wing politics had us replacing the explosives on our missles with concrete to "avoid civilian casualties" during The War On Terror, it's pretty obvious that if people found out about super-weapons there would be a political sh@tstorm about how nobody should be allowed to have stuff like whatever they are making.
A scientist inventing the "people eraser" for example doesn't want that kind of attention.
Oh sure, a lot of people would support this, but a lot wouldn't (our country is polarized). I still remember all the left wingers who went down to The Middle East to be volunteer human shields. That kind of thing combined with the adventures of Jane Fonda during Veitnam and other noteworthy examples are one of the reasons why I have such contempt for the movement.