What is the best flooring for my basement?
I lived in a house that had standing water once or twice a year.  It had a pad with berber(?) carpet on top.  It dried easy with a space heater and fan within a day of these wet periods.  Usually a 5x10 foot area got wet.   When I went to pull up the carpet both the carpet and pad were in perfect condition.  I asked a friend who had done installations for years and he said that I had an open-celled padding with no moisture barrier and if they are dried out thoroughly within a reasonable time he has never seen mold.

So new house I am in is basically about the same as yours.  I am putting down carpet in part of the basement with pad.  Almost all pads are open-celled and you would have to ask for one with a moisture barrier - so if you go to a carpet store and pick out their normal padding that is the right thing.

Also I am using Trafficmaster Allure Ultra on part of the basement which I have used on jobs.  I am not a HD homer - but I have used this stuff and it is awesome.  It is pure rubber and even if you got mold (you shouldn't) it would just clean right off.  

As for your $1000 budget.  You are going to have to get some very low grade carpet to meet that.  Do not get non-floating vinyl.  If you have to install the vinyl to the floor and you do have moisture problems you are in for a big mess.  The recommendation I have above is $1800.  Also a great website for flooring builddirect.com has other all rubber options but they may make you have a minimum order.