I assume the apartments in the Theatre District is Reserve Square? If so, those are very nice apartments, but like others have said, downtown is kind of difficult to live in (lack of things like grocery stores, etc.) The warehouse district is pretty nice as well -- the city's really tried to improve that area of downtown by building a lot of apartments and condos.
Where in downtown will you be working? Because if you're planning on walking to work (if you live downtown), because those two areas you've mentioned looking at apartments are considerably far apart (at least in walking distance).
Downtown traffic is not too bad, especially from areas like Lakewood (which is about as close to downtown you can get and still be considered in a "suburb"). I live about 18 miles outside of downtown and it takes me anywhere from 20-30 minutes to drive to work (I work downtown) on average. A lot of it depends on what highway you live closest to. I-90 and I-71 aren't too bad, traffic-wise, while I-77 can be a nightmare.
Lakewood's very nice, but it really depends on the part of Lakewood you're looking at (it's a fairly large suburb, and there's some "rough" sections). Lakewood has a lot of duplexes for rent, which are nice because you get a little more room than an apartment. There's a lot of young professionals that live in the Lakewood area. Brooklyn's pretty nice -- an older community, but there's a lot around it.
I could also recommend a few other areas, maybe a little farther outside of downtown that would be a little farther drive to downtown, but you might find some nicer apartments there.
Oh, forgot to add -- Welcome to Cleveland
