For though our act of sin, first, as to the subject that sins (man), and secondly, as to the act of sin itself (a sinful thought, word or deed that is soon gone), are finite: yet, if we consider sin: (1) In respect of the object against whom [it is committed] (the infinite God); (2) In respect to the absolute purity of God's law, a rule that bears out God's image set down by infinite wisdom, and that may be some way called infinitely pure; and sin, as being against this pure rule, that infinite wisdom has set down; (3) And if we consider it in respect of its nature, every sin being of this nature, that though it cannot properly wrong the majesty of God, yet as to the intention of the thing, and even of the sinner, it wrongs Him-sin in these respects may be called infinite. (Durham, James. Christ Crucified: The Marrow of the Gospel in 72 Sermons on Isaiah 53. Dallas: Napthali Press, 2007. Print. 224)Durham notes that both the act of sinning and the sinner himself are finite. The very worst sinner in the history of our race is finite and will pass away. The most grievous of sins are passed away immediately after the actual act.
However, in sinning we always sin first and foremost against God who is infinite. Also, sins are contrary to the infinite wisdom of God's law. And finally, though ultimately God's majesty is unapproachable, sins wrong His infinitely pure majesty. For these reasons, sins are essentially infinite.
This points to the reason why a just punishment for sinners is the infinite outpouring of God's wrath. It also points to why Christ's propitiation of god's wrath is efficacious; an infinitely pure Substitute took our place and paid our price.