I think North Campus (officially called Old North Columbus) is a great area. There seem to be many more grad students on this side of campus. East campus is very frat-heavy, south campus is kind of mixed but also probably undergrad-heavy. Old North is also really making a comeback as the nice neighborhood, because South campus is getting very shady very quickly. So there's lots of cafes and bars and a cupcake shop along high street north of Lane and south of Hudson-ish. Also a great up-and-coming neighborhood is Clintonville, which is just a bit north of Old North.
The other good area for grad students is Grandview, although it's not walking distance (and not near your lab). It's the area directly northwest of OSU campus (across the river) and there are LOTS of fun things to do in Grandview (bars, cafes, there's a candle-making shop where you can make candles).
Victorian Village is gorgeous but very expensive. Also, the periphery of VV is not very nice, and getting less-nice by the week. Same for Short North, and I wouldn't live south of King until approx 2nd-ish. That area in between has been really falling apart (boarded up houses, lots of creeps at night).