help needed for accomodation near manhattan,brooklyn ,ny

hi friends,
we will have to fly to manhattan in the month of july and we are thinking of taking an accomodation in manhattan or brooklyn.
manhattan is very pricey so i guess brooklyn will be a better bet.
we have a budget of 3000$/month
will that be sufficient to manage.
we have our 2 yr old son to take care of.
can anyone plz help us regarding accomodation info for rent ~800$....
also wat cud be the expenditure for the childs playgroup fees?
plzzzzzzz help
your advice is highly appreciated.
thanks a ton in advance
meg