I have in-laws who think this way about DC too and I would consider them well-adjusted and educated people. The fact is, DC is used as cannon fodder to prove points on right-wing news shows. And not just Fox--Sinclair owns the majority of local news channels across the country so they are getting these things shoved down their throats in between stories about the local little league team. If you have no experience with DC or live in a more rural area, then you are definitely going to think it's a crime-ridden wasteland. It sucks but it's true.

OP, I'm sorry you have anxiety about traveling to DC. It is highly unlikely that you will be the victim of a crime or even see a crime being committed (unless you count someone jumping over the metro fare turnstiles a serious crime). It's a great city with a lot of cool things to offer and I hope you take the chance to explore it with an open mind. If you are feeling really severe anxiety about visiting, find a conference buddy to do some touristy activities with and take Ubers instead of the metro. I do think you'll find the reality is much less scary than what you've been told by your family.