Here are two versions to help aid the question:
But to all who did receive him, who believed in his name, he gave the right to become children of God, John 1:12 ESV
But to all who have received him – those who believe in his name – he has given the right to become God’s children John 1:12 NET
The ESV identifies "all who did receive him" as those "who believed" (past tense). The NET identifies "all who have received him" as those "who believe" (present tense). Which reading more accurately reflects the underlying Greek language of John's gospel?
ὅσοι δὲ ἔλαβον αὐτόν, ἔδωκεν αὐτοῖς ἐξουσίαν τέκνα θεοῦ γενέσθαι, τοῖς πιστεύουσιν εἰς τὸ ὄνομα αὐτοῦ
ἔλαβον Is an aorist, active, plural, third person verb. It is the default tense when nothing really needs to be illuminated by the tense itself. It simply means that it happened at some point in time. In this case, "they received."
πιστεύουσιν Is a present participle which would imply present action. There are other nuanced forms of the present but for the sake of this exercise it is a present participle. Because it is articular, it seems to function more as a substantive in this case. Technically it should be "to the believing ones."
ἔδωκεν Is also aorist, third, singular. Literally, "he gave."
The only other verb-y thing in here is γενέσθαι which is an infinitive. It's translated as "to become" and functions as the content for what "he gave."
Aorist is a tense that used to be contested. It used to be perceived as perfective (past action that leads to present condition), but that's not defensible from the text and is usury supported by theology. That's backward.
To address your actual question, the past tense is what the underlying Greek supports.
In this case, the NET seems more interested in highlighting a theological reality and focuses on the present participle. This overlooks the fact that it is articular which would indicate that this participle is functioning more as a substantive noun (which is part of the function of participles) than as a verb.
EDIT: NET translation notes
On the use of the πιστεύω + εἰς (pisteuw + ei") construction in John: The verb πιστεύω occurs 98 times in John (compared to 11 times in Matthew, 14 times in Mark [including the longer ending], and 9 times in Luke). One of the unsolved mysteries is why the corresponding noun form πίστις (pistis) is never used at all. Many have held the noun was in use in some pre-Gnostic sects and this rendered it suspect for John. It might also be that for John, faith was an activity, something that men do (cf. W. Turner, “Believing and Everlasting Life – A Johannine Inquiry,” ExpTim 64 [1952/53]: 50-52). John uses πιστεύω in 4 major ways: (1) of believing facts, reports, etc., 12 times; (2) of believing people (or the scriptures), 19 times; (3) of believing “in” Christ” (πιστεύω + εἰς + acc.), 36 times; (4) used absolutely without any person or object specified, 30 times (the one remaining passage is 2:24, where Jesus refused to “trust” himself to certain individuals). Of these, the most significant is the use of πιστεύω with εἰς + accusative. It is not unlike the Pauline ἐν Χριστῷ (en Cristw) formula. Some have argued that this points to a Hebrew (more likely Aramaic) original behind the Fourth Gospel. But it probably indicates something else, as C. H. Dodd observed: “πιστεύειν with the dative so inevitably connoted simple credence, in the sense of an intellectual judgment, that the moral element of personal trust or reliance inherent in the Hebrew or Aramaic phrase – an element integral to the primitive Christian conception of faith in Christ – needed to be otherwise expressed” (The Interpretation of the Fourth Gospel, 183).
The NET translation seems to focus on the lemma πιστεύω. However, as with the SBLGNT which I used (for licensing purposes), the UBS4 and NA28 both have the articular participle. This is curious because the NET has the NA27 as it's primary Greek textual basis. I checked and the NA27 also has the participial form. In fact, even the WH Greek has the participle. As such, I believe that translations that focus on enforcing the verbal aspect of the word are (rightly) subjugating it to the main verb (ἔλαβον). A present participle that follows a governing aorist generally indicates action that occurs nearly simultaneously to the main verb. This is why, even though the participle is present, it is translated as past tense.Tweet