I’m a little biased because I live here, but I cannot recommend the Capitol Hill area enough. It’s very dog friendly, especially around Lincoln Park where everyone takes their pups to socialize after work. It’s also one of the safest neighborhoods in DC, although that depends to some extent on where you are (Capitol Hill covers a large area and the crime gets worse the further east you go). It might not be convenient if you work, say, in upper NW or Bethesda, but it’s easy to get to most parts of the city from here. Again, this depends to some extent on where you are, but most of Capitol Hill is near the blue/orange lines which go downtown, and the 90-series buses that go to U Street and Adams Morgan. It’s also very easy to hop on a highway from here if you have a car.
Capitol Hill is generally a quiet neighborhood that doesn’t have a lot of nightlife, but it’s definitely not sterile. Barracks Row has a lot of shops and restaurants, with new ones opening every month, and Eastern Market is a popular destination on weekends. If you do want nightlife, H Street is just north of Capitol Hill and a short walk away. Also, the 90-series buses that I mentioned earlier can take you directly to the other fun neighborhoods. Capitol Hill is a gorgeous neighborhood full of Victorian houses that are for the most part well kept and beautifully landscaped. And of course you have the Capitol building right there! It’s a great place to walk around.
Do you plan on renting or buying? I think the biggest challenge you’re going to have, if you’re renting, is finding a place that accepts two dogs (especially if they’re big). Again, Capitol Hill might be a good place to look in that regard; most of the rentals are English basements and the owners might be more flexible with allowing dogs.