No, you don't want to replace (or substitue) 'with' with 'about' because you have a comparison here with 'better'. The complete sentence would be sth like 'I should have known better than to fool around or get interested in or fall in love or get involved with a girl like you. Therefore, 'than to do sth' is understood and leaving it out makes it easier to keep the rhythm. Yes, it's fairly common in some circles to leave out the 'than to do sth' and just say I should have known better with them or simply 'with her, I should have know better.'
To know about is different in your example because there is no comparison. I should have known (more) about her before asking her out. I should have known better than to go out with a girl like her. You should know more about her before you go too far.