I'm an undergrad student starting graduate school at OSU and I opted to move to the Grandview area for the next school year. Grandview is a nice, smaller suburban area of Columbus where a lot of young professionals live due to the closeness to the medical center and campus. It is definitely also MUCH quieter than off campus.
I would also suggest not to live east of High Street in the off campus area. The landlords in this area do a lot of shady business practices. I've lived directly off of High for 3 years in 2 different places and there have been this year alone 2 break ins in my apartment complex, theft and burglary in the surrounding houses, random people sleeping on my stairwell at night, etc. I would not suggest living alone in these areas, especially towards south campus unless you install a security system.
Other areas that a lot of graduate/phd students live are in Clintonville and Victorian village. I've worked in Clintonville for two years and while I think that it is safer than the off campus area, it still has a small amount of crime from time to time. The only issues I would see in moving to Grandview would be the higher cost as well as the need to have a car and buy a parking pass. In Clintonville, COTA busses can take you directly to campus.