I'm a graduating senior undergrad at OSU. There are some nice high rise apartment buildings on Lane north of campus, and I think that many law students live in apartments south of the law school in Short North. I'm not really sure how those are or how they find them.
I live in University Village, an apartment complex about a 3-5 minute drive north of campus, and I absolutely love it. The rooms aren't huge, but the quality is nice and relatively cheap as far as how things go around here. Some law students live here as well, so I'd at least check it out. It's safe.
Good luck! If you have any more detailed questions, let me know.