I would have been a horrible law student at 23 (note that I'm not saying that 23-year-olds are terrible law students - just that *I* would have been.)
I was worried about this too, so I 1) talked with people from the mature students' club at my school 2) talked to the admissions office, who assured me that about 20% of the incoming 1Ls have graduate degrees (so do I) and are 27 or over 3) met other older students at the admitted students' welcome. Turns out a friend of mine from grad school, now in her 40s is going to my school too (which tells you something about the academic job market, doesn't it?) and 4) I talked to a BigLaw partner that I happened to have a connection with. He said that his firm, at least, valued the maturity and varied experience that a lot of older students bring and they've had a great deal of success hiring non-trads. His advice for me was the same as it would be for any law student: go to the best school that you can, work hard, get good grades, and the rest will follow.
And yeah, if some firm won't hire me because I'll be 37 when I graduate, that firm would obviously be a bad fit anyway.