Her brother does not like me...

This is about my girlfriend’s brother, who is older than she is and seems to have a problem with me. Whenever I go to visit her, he makes it a point to make me seem inferior, like I don’t have the right to date his sister. He keeps telling her she can do better, which is insulting but also worrying because she looks up to him and I am constantly afraid that she may one day take his advice. I have tried chatting with him and am always nice and polite to him, but nothing I say or do changes his opinion of me or makes him behave any better. How do I deal with this?
You have dealt with it in the way all mature adults should, by being respectful and polite. He has not extended that same courtesy towards you, which reflects poorly on him. I suggest you tell your girlfriend why this bothers you and acknowledge that she is with you because she has a mind of her own. You can’t expect everyone to see you the way she does, but you can do everything you can to focus on your relationship with her and make sure the two of you are constantly growing together. Communicate, be honest, and trust in the fact that she will do what is right for her. If she chooses to listen to her brother, that is her prerogative, but it doesn’t make sense to assume she will because it implies an inability on her part to think for herself.

Should I tell my boyfriend that I don’t like the way he dresses? I am sometimes embarrassed to be seen with him and I know that makes me feel shallow. Should I just accept him the way he is and try to cope with his bad taste?
Taste is subjective. If you don’t like something, you are free to air that opinion, but he is also free to accept or reject it. Accepting someone for who they are always matters.