There are really a lot of great neighborhoods close to downtown. Since you didn't mention what kind of "budget" you were on, I am not taking price into consideration.
One of my favorite neighborhoods that has come on in the last few years is the Uptown neighborhood. It is minutes from downtown and has quite a bit of new and renovated apartments and homes. A friend of mine leaves in the western edge of Uptown in the San Rafael Neighborhood and loves it! http://www.neighborhoodlink.com/denver/sanrafael/
Also, the Whittier Neighborhood has had quite a bit of cool renovation in the last few years. http://www.neighborhoodlink.com/denver/whittr/
In fact a friend of mine recently renovated a old garage into 4 lofts. I think they were for sale, but they might also be for rent. They are at 26th and Franklin. http://www.freeformlofts.com/
The 80205 zip code encompasses a lot of neighborhoods in and around downtown and you can find a lot of info on this page http://www.neighborhoodlink.com/zip/80205
Also, if money really is not a problem, the Country Club neighborhood is 5 minutes from dowtown right off of Speer. It is lovely, but I doubt there is a lot to rent in that area. http://www.countryclubhistoric.org
Also for pictures, schools, and more links: http://www.neighborhoodlink.com/places/Denver_CO