Interestingly enough from a text critical perspective the support for both readings is strong, and both can be justified from context.
20:31 πιστεύ[σ]ητε (that you may come to believe) {C}
Both the present tense πιστεύητε and the aorist tense πιστεύσητε have notable early support. The aorist tense, strictly interpreted, suggests that the Fourth Gospel was addressed to non-Christians so that they might come to believe that Jesus is the Messiah. The present tense suggests that the aim of the writer was to strengthen the faith of those who already believe (“that you may continue to believe”). Since it is difficult to decide between the readings by considering the supposed purpose of the writer, both readings are included by putting the letter sigma in brackets. Whether the distinction between the present subjunctive and the aorist subjunctive here can be maintained is debated. Carson (The Gospel According to John, p. 662), for example, writes that “it can easily be shown that John elsewhere in his Gospel can use either tense to refer to both coming to faith and continuing in the faith.”
Translators, however, may have to choose between the two readings. NRSV follows the aorist tense: “so that you may come to believe,” and states in a footnote “Other ancient authorities read may continue to believe.” It is not clear which reading is the basis for the REB translation “in order that you may believe that Jesus is the Christ.” The footnote in REB states: “witnesses read different tenses, some implying continue to believe, others come to believe.”
Omanson, R. L., & Metzger, B. M. (2006). A Textual Guide to the Greek New Testament: an adaptation of Bruce M. Metzger’s Textual commentary for the needs of translators (pp. 211–212). Stuttgart: Deutsche Bibelgesellschaft.