In the US it is probably easier to list the ones that have rail (heavy or metro) links:
Atlanta Hartsfield - MARTA Red Line rapid transit
Baltimore - BWI has an Amtrak/MARC station, also light rail link
Boston - sort of, need a shuttle bus to get to the MBTA Blue Line station
Chicago - both airports connected to rapid transit (the "L")
Cleveland - red line, one of the first in the US
Dallas/Fort Worth (DFW) - DART Light Rail
Denver
Los Angeles - under construction
Newark NJ - people mover connection to Amtrak/NJT Northeast Corridor
New York - JFK yes, LaGuardia a resounding no
Philadelphia - SEPTA Regional Rail
Providence RI (TF Green Airport) - Station on Amtrak Northeast Corridor but only commuter rail (MBTA/RIPTA) stops there
San Francisco - BART system
St. Louis - light rail
Seattle (Sea-Tac Airport) - light rail
Washington DC (both Reagan National and now Dulles) - WMATA rapid transit
I have probably missed a few
New York City John Fitzgerald Kennedy airport and New York City Newark airport are both connected solely by rip off monorail things. So it is debatable whether they count. You have to take the monorail which then takes you to a railway station outside of the airport rather than it being an actual airport railway station.
Phoenix Arizona is similar to the above ones. It has a monorail which takes you to a tram stop and then you can take a tram from there in to the city centre. But at least the monorail to the tram stop is free of charge here.
Portland Oregon has a tram stop outside the airport terminal with frequent trams in to the city centre.
Salt Lake City Utah has a tram stop outside the airport terminal with frequent trams in to the city centre.
Minneapolis Minnesota has tram stops outside both of the airport terminals with frequent trams in to the city centre.
Los Angeles Burbank has two railway stations on two different Metrolink suburban train lines within a short walking distance. Burbank Airport South is the closest and is only a five minute walk to the terminal. Burbank Airport North is slightly further and takes about fifteen minutes to walk to the terminal.
Baltimore Martin State Airport has a dedicated railway station on the MARC suburban trains a short walk from the terminal. But the airport is only used by private aeroplanes now so not much use. Still i suppose it is helpful for airport workers.
Albuquerque New Mexico has an airport railway station that is laughably named after the airport despite being a one hour walk away. There is a free bus (222) but it is mainly for airport workers as it only runs twice a day in each direction. Therefore it only connects with a very small amount of the flights. So despite its name i would not classify this as an airport railway station. In fact it is far easier to simply take a free bus (50) in to Albuquerque city centre directly as that bus runs every thirty minutes.
South Bend airport has a dedicated railway station directly outside the terminal building. It only runs 9 times a day on weekdays or 6 times a day on weekends so it is not exactly a frequent service. About one train every couple of hours.
Chicago O Hare additionally has a monorail connecting it to a mainline station on the Metra suburban trains. The monorail is free of charge. But as you say the airport is also served by the Chicago L Metro trains.
Hartford Connecticut has a short shuttle bus ride to Windsor Locks station which has mainline trains.
New York City Stewart has a short shuttle bus ride to Beacon station which has mainline trains.
Miami Florida has a mainline station and metro station. The airport has both Metrorail urban metro trains and Tri Rail suburban mainline trains.
Fort Lauderdale Florida has a mainline station named after the airport. But this requires a free shuttle bus as it is quite a distance from the terminal.
Orlando Florida has a mainline station but it is only served by Brightline long distance trains. There is no connection to the local Sun Rail suburban train network and no trains to Orlando city centre.
Milwaukee Wisconsin has a mainline station named after the airport. But this requires a free shuttle bus as it is quite a distance from the terminal.