When it comes to dealing with the word βαπτίζω, I find Vos's discussion in his
Reformed Dogmatics to be most helpful. I know this is a lengthy quote, but it is well worth the read:
11. What word in the New Testament is used for baptizing, and what is the force of that word?
Βάπτω (baptō) means "to dip," "immerse," for to baptize and to immerse are related to each other (cf. the English "to dive," "to dip"). In John 13:26 it is used for the dipping of the morsel by Jesus that he gave to Judas. See also Ruth 2:14 [Septuagint] for dipping the morsel in vinegar by Ruth; Luke 16:24 for dipping Lazarus' finger in water. In all these passages, the word used is βάπτω, so that positively no doubt need exist about the original meaning. When a Baptist says that baptō means "immerse," then one should grant him that without reservation.
A strengthened form of baptō is baptizō (βαπτίζω), and this is the usual word in use for "baptize." This, too, is originally "immerse." Actually, baptizein means "immerse repeatedly"; it is an iterative form. Hence it is used instead of baptō in 2 Kings 5:14 [Septuagint], which recounts how Naaman dipped himself seven times in the Jordan. But this iterative meaning is not always maintained, so that often enough baptizein is equal in meaning to baptein. Now, however, this baptizein appears at the same time to have been the usual Greek translation for the Levitical washings and purifications, which again may be connected with the V 5, p 122 fact that this took place generally by bathing—that is, by immersing the body in water. One may compare, for example, Matthew 15:2, "For they do not wash their hands when they eat" (οὐ γὰρ νίπτονται τὰς χεῖρας), with Mark 7:4, "unless they first wash" (ἐὰν μὴ βαπτίσωνται).
With this, the concept of baptizō is placed directly under the viewpoint of "washing," a fact of the highest importance. That this "washing" was in most instances a washing by immersion appears as something accidental that could also be something else, and that, if it had been something else, nothing of substance would have changed. The Levitical purifications were washings of the whole body, also where a sprinkling accompanied them (Num 19:19; see also v. 18; Lev 11:24–28 and following; 17:15; 14:2–8; 15:16–18, 19–24, 25–29, 2–15). That in all these purifications the whole body had to be washed and not just a part had to do with the ceremonial and burdensome character of the Old Testament dispensation. If an easy sprinkling had been sufficient, then perhaps it would have quickly become an outward form. The bathing of the whole body did not easily become a meaningless custom. Then one should also not forget that for the Levitical purification its symbolic character came to the fore. It was therefore in all respects fitting that in the case of uncleanness the whole body was subject to a washing. By that was certainly pictured how the entire person is polluted by sin and how complete renewal is necessary. However, from that it may not be inferred that at baptism as a sacrament, too, complete immersion of the body is necessary. With baptism, not the sign but the seal is surely in the foreground. The New Testament sacraments are not in the first place symbols; they are above all seals of the covenant. Thus it is in no way necessary that the entire symbolism of the purifications of the Old Testament be transferred to the baptism of the New Testament.
The error the Baptists make when they insist that baptizein is immersion and nothing else lies in overlooking the fact just mentioned. Words have their meaning by their use, not by their etymology. One can safely grant not only that originally baptizō means to immerse; indeed, one can even go so far as to say that initially immersion was the customary mode of baptism, without playing into the hands of the Baptists. The point at issue between them and us surely lies in this: whether immersion constitutes the heart and essence of the symbolism of baptism, so that abandoning it would be the same as abandoning baptism itself. When one asks a Baptist, "Why did Christ institute the sacrament of incorporation into the Christian church in this way?" then his answer is: "Because it had to be portrayed by descending into and emerging from the water." That thereby washing takes place at the same time, since one cannot immerse someone without the water at the same time washing his body is, according to him, something incidental. Baptism would be baptism, and its essence preserved, if one could immerse someone in something else that does not have a cleansing quality. If one poses the same question to us, then we answer: The sacrament was instituted by Christ in this way because He intended to have washing and purification portrayed. The fact that this ordinarily took place in a land like Palestine and according to the Jewish law by immersion or bathing was something incidental and subordinate. If a washing takes place without immersion, then baptism retains its essence.
Thus the issue between us and Baptists is not at all whether baptizein means to immerse or to sprinkle. One can grant, and probably will have to grant, that nowhere in the New Testament has it completely lost its original meaning of "immersing" or "dipping." The issue is simply whether immersion was the main point or something incidental. And then we say the latter. It was immersion with the purpose of washing, and in order to portray purification. We rely on this when we claim that baptism by sprinkling is just as much the ordinance of Christ as baptism by immersion. From their side, Baptists believe that the Reformation has taken half measures, that Luther and Calvin did not fully clean out the Roman Catholic leaven, and that on them the duty rests to restore original Christianity in its purity.
One should preferably not combat Baptists with weak historical arguments from the New Testament. One can appeal to Mark 7:4: βαπτισμοὺς ποτηρίων καὶ ξεστῶν καὶ χαλκίων καὶ κλινῶν, "washing of cups and pots and copper vessels and couches." It is said that cups and pots and copper vessels could be immersed, but not couches. It is a question, however, whether these words (kai klinōn) belong in the text. Westcott and Hort omit them. The Revised Version does also. One can also point to Acts 2:41–42. There were 3,000 people added to the church, who for the most part were also certainly baptized. Was that possible in so short a time if baptism took place by immersion? It is not impossible. An equally large number of converts have been baptized in a relatively V 5, p 124 short period of time by immersion. The appeal to Acts 10:47 is also not strong, since we evidently have to do there with a figurative expression. Peter intends to say: the Holy Spirit has already come upon them; can anyone still forbid water, by which they are signified and sealed? From Acts 16:33 it has generally been inferred that the jailer and his family were not baptized by immersion but simply by sprinkling. But Baptists say that the jail, like most of the large buildings in the Middle East, had a fountain and a cistern. First Corinthians 10:1–2 states that all the fathers were under the cloud and passed through the sea and were baptized into Moses in the cloud and in the sea. The fact that they went through the water, and that certainly with dry feet, can, it is said, be called already being baptized. They were simply sprinkled with the spattering drops. But there is no mention of sprinkling in the account of these events. The apostle apparently conceived of them such that the sea and the cloud surrounded the people, and so became an element in which they were located. The sea was on both sides, the cloud was over the Israelites; that was their baptism.
Appeal is also made to passages that speak of a baptizing with the Holy Spirit: Matthew 3:1 (ἐν πνεύματι ἁγίῳ καὶ πυρί); Mark 1:8 (ἐγὼ ἐβάπτισα ὑμᾶς ὕδατι, αὐτὸς δὲ βαπτίσει ὑμᾶς ἐν πνεύματι ἁγίῳ); John 1:33 (οὗτός ἐστιν ὁ βαπτίζων ἐν πνεύματι ἁγίῳ); Acts 1:5; 11:16 (ὑμεῖς δὲ βαπτισθήσεσθε ἐν πνεύματι ἁγίῳ); 1 Cor 12:13 (καὶ γὰρ ἐν ἑνὶ πνεύματι ἡμεῖς πάντες εἰς ἓν σῶμα ἐβαπτίσθημεν). The question is how this "in the Holy Spirit," as it literally stands in all these passages, is meant. Is the Holy Spirit the element in which one is baptized, in which God, as it were, immerses, or is it to be understood as a being baptized with the Holy Spirit? In the latter case, ἐν, equivalent to בְּ in Hebrew, would be an instrumental preposition. In Mark 1:8 it is without doubt "by means of water … by means of the Holy Spirit." But from this it does not yet follow that sprinkling is thought of, for one can also call baptism through immersion a baptism by means of water. In 1 Corinthians 12:13, the translation "by means of one Spirit" appears to us the most natural, but sprinkling is not proven by it. By all of these things one can only deprive Baptists of proofs, not obtain proofs for his own views. See the commentaries on these passages.
—Geerhardus Vos, Ecclesiology, ed. Kim Batteau and Allan Janssen, trans. Richard B. Gaffin, vol. 5, Reformed Dogmatics (Bellingham, WA: Lexham Press, 2016), 121-124; all italics original.