Stick to the red line of the
Metro to get yourself to Rockville easily. Woodley Park is very nice, Cleveland Park and Van Ness are good too. Dupont Circle is more urban, but pricier. All of these areas are very very safe. I wouldn't go any further down on the red line than that, it just gets too far for Rockville.
If you mean a studio or 1-bed for $2000, that's a good budget. If you mean for a 2-bedroom, look in Maryland.
As far as specific streets, if you stick to near those metro stations, you'll be fine. Otherwise, if you have a question about a particular area, you can post back on here. That side of the city is very safe, so really the main concern would be where the metro is and what's nearby (all available on GoogleMaps).