Dupont Circle is a great and vibrant, clean location. I definitely feel safe, even late at night, but the noise level can get very bad, especially on the weekends.
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You have no more of a chance of being a victim of a crime on Metro than you would on street level. I absolutely think you will be perfectly safe on metro any time of day. Don't waste your money on a taxi.
There is a reliable Washington DC police presence, especially around the main tourist areas, and violent crimes are generally very low. As a solo traveler, you'll just need to do your research to avoid dangerous areas and take extra caution when making your way through the city after dark.
The public may visit the sites of National Mall and Memorial Parks 24 hours a day, with the exception of the Washington Monument which is open 9 a.m. to 5 p.m. Rangers are on duty at the sites to answer questions from 9:30 a.m. to 8 p.m. in November - February.