With Google Mail (and Apps) anything after a + in the first part of the address is ignored, so foo+dropbox@gmail.com would be routed to foo@gmail.com. That's the easiest way to do it that I know of. No need for managing separate aliases.
Another Google Mail trick is to use periods. Not as useful as the +, but for those sites that don't accept +, one can usually add in a few extra periods to place sites into buckets (multiple adjacent periods don't work).
m.y.e.m.a.i.l@example.com
You probably just locked the stranger out of accessing their account though, so you probably shouldn't do this, unless said stranger is signing up for all kinds of services using your email address, in which case maybe they deserve it. :p
Plus you can't completely shut down a label. You could route it into the trash but it will still end up in your email account.