I don't live in the Capitol Hill area but I do work very close to the Navy Yard Metro station. I also sometimes take the N22 Metrobus between the Navy Yard area and Union Station. Admittedly, I don't have that good of a time perspective because I just moved to the area/started my new job about 1 1/2 months ago.
The Navy Yard area is going through some major transition. Some newer development, particularly at the western end of the "neighborhood" near South Capitol Street (with the Washington Nationals' new ballpark being the most notable development but also including a hotel, some currently under construction condo units, and the U.S. DOT headquarters) is modifying the area dramatically. This general area still has some very rough locations however; in general I'd say the locations closest to the south side of the Southeast Freeway are the roughest. As an example, there are still some very run down attached dwellings on 3rd Street SE just south of the freeway and north of the DOT buildings (and across the street from some of the designated ballpark parking lots). The 8th Street SE commercial corridor does extend south of the freeway down to M Street/northern edge of the Navy Yard, but there is some rough development near the corridor (and in a couple spots, along the corridor). The adult entertainment clubs that I've been told were in the general Navy Yard Metro station area are long gone however.
When I accepted my new job in DC, I actually looked into this area, seeing I could walk to work, it is close to a Metro station, and that it also has good access to I-295/I-395. However, IMO there isn't really enough development there yet to be attractive to most people. (I do like eating at the Five Guys on 2nd Street SE between L and M Streets though

.) On the other hand, the DC government is focusing more on this area for redevelopment opportunities than on any other area in the entire city, and there could be some dramatic changes to the neighborhood in the next 2-4 years if most of the proposals pan out. National and regional market trends will probably determine how many proposals do get implemented in the proposed time frame.
Oh, in terms of walking between Capitol Hill/Eastern Market and the Navy Yard area, during the daytime there shouldn't be any problems, and the same would probably be true on nights when the Washington Nationals are playing a home game. Other nights however might be a little sketchy at this point. FWIW, I know there are underpasses on 3rd, 4th, and 8th Streets (and there may be a few others too east of South Capitol Street).
Incidentally, I think the area just north of the Southeast Freeway, especially the further west you go closer to the U.S. Capitol Building, is very nice, just from passing through there (both in vehicles and once on foot).