If you want to do a summer internship in the Middle Atlantic states, being a UVa student will be a big plus, because UVa is respected and has an extremely supportive alumni network. You are correct that internships during the school year will involve more varied opportunities in a big city than in Charlottesville.
Charlottesville is a very safe city, with a few occasional exceptions.
Keep in mind that New York City is extremely expensive - particularly for housing, restaurants and food. Even items in drug stores cost much more. Charlottesville is a very affordable city.
DC is a 2 hour drive from Cville. There is daily train service, plus a bus that travels from UVa to the DC metro system every weekend. There also is regular commercial bus service, plus lots of people willing to carpool.
One of the nice things about Cville is you are accessible to visit DC, Baltimore, Philadelphia, NYC and Boston (particularly by train), without having to pay the prices for living in a big city.
I love visiting New York City, but I come home exhausted. If you are already trying to adjust to the stress of college and living in a new country, adjusting to the stress of NYC might be a bit much.