At the beginning of the purport in the Srimad Bhagavatam 1.6.21, Srila Prabhupada talks about not being able to approach Krishna if there is any tinge of material affection. I guess the question would be, what does he mean by approach? We start our devotional service as conditioned souls, and I take devotional service to be an approach towards Krishna.

Also, he says devotional service begins when one is freed from passion and ignorance. This means one has to at least be in the mode of goodness from which we approach Krishna in devotional service. Yet, Arjuna, a pure devotee, is a ksatriya and is in the mode of passion, correct?

Your understanding is correct. Prabhupada is using ‘approach’ here in a special sense, of actually coming directly into the association of Krishna with full consciousness. The general rule is ‘ye yatha mam prapadyante…’ As I often explain, the verb prapad [yante] literally means ‘approach.’ In English, approach can mean ‘to come near’ or ‘to come nearer.’ Generally, we use the later definition, but Prabhupada, in this special case, uses the former.