If I wanted to say: She told me she knew my brother.
Would this be correct? Or would I have to use the passive form?
Both sentences are grammatically and semantically correct, but as Choco mentions in the comment below, it's more natural to leave out the 僕の in 僕の弟, because 弟 without a qualifier such as ～さん or Xの～ is always taken to mean the speaker's own younger brother.
Other than that, which form is more natural depends on what aspect you want to emphasize. Without knowing the context the phrasing most likely to be used by a Japanese person would simply be:
彼女は弟を知っている [と言いました / と言った / って]。
However, you've now lost the explicit information that tells the listener that it was you who she told she knows your brother.
If that information is important, there are two ways to rephrase that sentence:
You can add 僕に as you did in your 1st sentence. Keep in mind though that doing so can add more emphasis on 僕に than natural given the context (especially when you use the word order you did where 僕に is placed close to the verb).
You can use the passive form as you did with your 2nd sentence. This is done very often because it has the benefit of being grammatically unambiguous without adding unwanted emphasis on the you-part: