I get why your family has this view of LA and the area around DL. It is very understandable, I would might not want to go back to a place with that negative association for me either.
But I agree with those who say it is not that bad around DL. Bad experiences can happen everywhere. Shootings happen all over DC, even places that are considered relatively safe. There has been at least one shooting and stabbing at the National Zoo. Heck, a Congresswoman was mugged just last night. Problems can happen anywehre, particularly in major cities.
...said the girl from Baltimore.
(I know, you are from the County, not the city itself).
I would consider B'more waaaaaaay more dangerous. Even the formerly "safe" areas like Inner Harbor. I would be far more concerned about going there than DL. There was a time I went to Little Italy quite a bit.