The following quote is from the introduction of Opening the Gospel of John, by Philip W. Comfort and Wendell C. Hawley, page XVIII.
[John’s] Gospel was written primarily to those who already believed yet needed their faith infused with a fresh breath of life and strengthened by a clear presentation of Jesus Christ, the Son of God. (The verb tense for believe, a present subjunctive in the earliest manuscripts containing John 20:31, indicates that John wrote this Gospel to encourage ongoing faith, more than to produce initial faith, though it can certainly do the latter quite well. The other verbal expressions in John 20:31 — “believing” and “have life” — are also in the present tense, emphasizing continual action.)