The National Mall is very heavily policed and very safe. Muggings have occurred at night and received major press coverage due to their sensational nature, but they are extremely rare. Use the most basic common sense and you're guaranteed a lovely evening late into the night. Note, though, that the fountains are turned off at midnight, so it is better to see the sights beforehand.
contact
The Open Park Project is working to make the whole Mall between 4th and 14th St one big wireless hotspot, but until then you'll have to head north to Freedom Plaza 14th & Pennsylvania Ave or east to the steps of the Supreme Court and the Library of Congress for your free wireless needs.
Alternatively, the Smithsonian Castle 10th St & Jefferson Ave SW provides free WiFi in the immediate vicinity, and its gardens are a perfectly lovely place to catch up on your email.