Old Town Alexandria is accessible from the Washington, DC Metro system via the King Street-Old Town station on the yellow line. See here for a Metro map and here for more information on the King Street station.
The closest airport is Reagan National. Reagan National is on the Metro yellow line, two stations away from the King Street station. The DC area is also served by the Dulles and Baltimore-Washington International airports, but each is much farther away from Alexandria.
King Street Trolley
There is a free trolley that runs from the King Street Metro station to the waterfront, passing very close to the Tavern (at the stop on the corner of King Street and N. Royal Street). It runs daily every 15 minutes. See the website here for more information.
There are several parking facilities serving Old Town Alexandria, and most garage parking is $5 per day on weekends. The most convenient to Gadsby's are the Tavern Square Parking Garage, 418 Cameron Street, which has a pedestrian entrance in the same block of Royal Street as the Tavern, and the Market Square Parking Garage, 108 N. Fairfax Street, which is directly across Royal Street from the Tavern. Complete information on parking is available here.
There is also metered parking on a number of streets.
If those staying in a particular hotel want to, they could also share a taxi to the Tavern. See Alexandria Yellow Cab for example. Uber and Lyft are also common in the DC area.