The nicest area of Brooklyn I went to was Carroll Gardens which has a realy nice neighbourhoody vide going on, not to mention some good restaurants. Could be worth a go.
I stayed in Williamsburg on South 2nd Street. It's a tad loud down that end but plenty going on.
I must say the amount of tourists on Brooklyn Bridge kind of detracts from the enjoyment. Some great views over in Brooklyn, especially from the Wythe Hotel bar terrace.