There are a couple of airlines (Qantas, Finnair, etc.) that do not arrive every day. Is there any airline in particular you're looking for? If not, I'd say that any day is really fine. Most of us tend to only go on the weekends because that's all the free time we have.
If you're looking for Qantas, Wednesday is a good day! :) Qantas usually arrives between 4:45pm and 5:30pm. You gotta be careful if JFK is using 22L/13L for arrivals, because while Qantas would usually arrive on 13L in that situation, Brian, Eric and I learned the hard way that it is always possible that Qantas will go to 22L. Last week when we had a tip that Wunala Dreaming was coming, we expected an arrival on the 13s since Qantas is coming from the west, but we were of course prepared for the possibility that ATC would screw us over by vectoring Qantas to 22L. We tuned in to New York Approach, and when Qantas checked in, sure enough it was told to proceed to 22L. We figured we had about 15 minutes before it would be on final, so we high-tailed it to the Mounds for the 22L arrival (probably breaking the speed limit on every road we were on) and made it with about 30 seconds to spare before Wunala Dreaming coasted in. It was awesome.
Well, it looks like I am going down on Thursday because my mom has to pick up a friend coming on the British Airways flight from Heathrow at 3:50 but she said she'll take me at 12 so I could go spotting.
Depends on what runways are in use. If it is in fact 22L and 13L for arrivals (it wouldn't be 22L and 13R except in unusual circumstances), then my money is on Cathay landing on 13L but Emirates landing on 22L. Here's a shot of mine from a couple of weeks ago of the Cathay A346 arriving on 13L: http://www.airliners.net/open.file/901128/L/. Don't hold me to this, though, because weird things have been known to happen (i.e., Brian, Eric and I narrowly avoided heartbreak when "Wunala Dreaming" was unexpectedly put on 22L instead of 13L).
Your best bet is to hope that 31R is the active arrival runway, because then both Cathay and Emirates will likely arrive on the same runway.