Well, no. The Constitution is not the be-all and end-all of governance, nor does it forbid all possible onerous behaviours. The position of the USG is that what the NSA engages in is constitutional. Even if that were the case, that wouldn't make it a good idea--unless it were necessary. You could implement a 1000% tax on beer and it would be perfectly legal and constitutional, but would also need to be somehow necessary in order for it to be justifiable.
Even if what the NSA is doing is deemed constitutional, that doesn't mean it's right or a good idea. In order for that to be the case, it'd also have to have a strong claim of necessity.