If my brother wants to open a LLC under my name, what are the some of the major drawbacks or risk associated with it?

If you are a member (owner) of the LLC, you not generally liable or resonsible for any debts of or judgments against the LLC except:
1) if you personally guaranty anything (like a lease or loan);
2) for certain tax (e.g. income tax withholding) debts, if you are the responsible person for making sure they are paid;
3) if you comingle personal and business funds, or otherwise act in such a way that blurs the line between you and the LLC--this can let others "pierce the corporate veil" and negate the liability protection LLCs generally provide.
Except for the above, you should generally have good protection from liability. However, note that if your brother does anything improper with the LLC or its funds (e.g. takes customer/client money; pockets tax payments; etc.) and you are aware of that wrongful conduct but do nothing about it and in fact essentially aid him by concealing the bad conduct, you could face both criminal and civil liabilty.
Question for you to consider: why does your brother want to open an LLC under your name and get you involved in this way? Why can't he do what is more logical and intuitive, and open his own LLC? If the issue is that he does not want to be named or associated with it, or his own name, due to what he's done in the past, is bad, you probably don't want to get involved.