It could literally mean to grasp someone's hand or to handhold and guide someone on a learning journey so that they will have an easier time. More context is needed here.
"for" goes with "a dollar" as a prepositional phrase to provide more information on "I will not hold your hand". Under what circumstance will I not hold your hand? Even if you give me a dollar.
"for a dollar" reminds me of a YouTube clip where an ugly man pretends to approach women on the streets, asking for kisses. When they refused, he jokingly said, "I've got a dollar!" which suggests that it's easy money for an easy task. Again, more context is needed here.
If the writer wanted to refuse someone's request to hold their hand under any circumstance, the writer would probably use "for a million dollars/bucks". (No, I will not hold your hand for a million bucks.")
I hope this helps.