There is no inexpensive way to do this from DCA. Sure you can take the metro to Union Station and then take Amtrak up to BWI. The Maryland Commuter trains don't run at night.....so they're still useless.
There may be a bus but I wouldn't take a but that late at night with two kids. If you're from LI, why not fly WN from ISP? You can fly for $39 per person to BWI the night before.