Early depictions of the Passion of Christ tended to omit the crucifixion, and there are very few representations of it in the Early Christian and Early Byzantine period. The Basque Crucifixion was shown to be a modern fake created by Basque separatists; who's represented in the Alexamenos Graffito will never be certain, nor can it be dated with much certainty; and the fresco in a tomb on the Esquiline pre-dates the Christian period by several centuries. Several graffiti from Pompeii mention crucifixion but as a Roman insult or punishment (source). Although the Romans used crucifixion regularly, again in pre-Christian art it was very rarely depicted. There are very few images of crucifixion, and not all can be linked to Jesus' crucifixion.
In Early Christian art they went out of their way not to depict it. So for example in the circa AD 400 century mosaic in the apse of Santa Pudenziana (here), a jewelled cross is shown at Golgotha, not the crucifixion. In fact, there is rather little evidence for the use of a cross as a symbol by Christians before the time of Constantine. On early Christian sarcophagi, such as the sarcophagus of Junius Bassus (dated by inscription to AD 359) or the Dogmatic Sarcophagus, scenes from The Passion were represented - just not the episode of the crucifixion.
The solution fond by the designer of the mid-4th century Passion Sarcophagus (Vatican, image) was to replace the crucifixion with the Chi-Rho:
This ivory plaque is one of four made to decorate a reliquary around AD 420 (The Maskell Ivories, British Museum). It's the earliest depiction of the crucifixion which depicts Jesus for certain, as indicated by the inscription on the titulus crucis: REX IUD[AEORUM]
The doors of Santa Sabina on the Aventine in Rome show one of the few public depictions of the crucifixion. The doors seem to be original to the church, which was constructed by Celestine I (AD 422-433) according to its inscription.
Like many early crucifixions, Jesus and the thieves are shown in the orans pose, their arms outstretched and palms up, a pose associated with prayer - the orans pose was not new to Christian iconography, but seems to originate in popularity with images of Artemisia II of Halicarnassus, which were in turn used as a type for depictions of Roman empresses. The figures are all shown 'standing' on the base of the panel, and the crosses are barely visible, so the fact that they were being crucified was not emphasised. This orans or orant pose is also described in sources as having been used in prayers, such as a letter describing the martyrdom of Blandina, as being a deliberate emulation of Jesus' pose on the cross.
There were no earlier images of the crucifixion to serves as examples, but Sheckler and Leith in their study of the Santa Sabina doors suggested as a prototype depictions of the Three Boys in the Fiery Furnace (Daniel 3). They give the example from the Catacombs of Priscilla, where several popes were buried, including Celestine who constructed Santa Sabina:
And it is also depicted on the Passion Sarcophagus mentioned above:
Although scholars like to create Darwinian evolutions of iconography, it's interesting to note that in the contemporary ivory and doors Jesus is shown with and without a beard. I also wanted to show that the orans pose was used to show deification, for example on this coin Domitian issued on the death of his infant son:
The next images of the crucifixion that survive are not until almost two centuries later. The inside of the lid of a 6th century box from the Sancta Sanctorum of St John Lateran that held relics from the Holy Land (now in the Museo Cristiano, Vatican):
Also from the Holy Land are a whole series of ampullae from the altar at Monza Cathedral and S. Colombano in Bobbio, given by Queen Theodelina of the Lombards in AD 603 and AD 613 respectively (Monza below):
On the vast majority of these Jesus is shown in a bust above a naked cross. There is a similar depiction for example in the 520s mosaic in S Stefano Rotondo in Rome, where two saints flank a cross symbolising the crucifixion, but the actual crucifixion is not depicted. The ampullae can be dated by the gift of Theodelina and by the fact that the Holy Sepulcher they depict was destroyed by Chosroes II in AD 614. The most common ampullae, such as this one of c AD 600 in Dumbarton Oaks, show the two thieves in the orans pose, flanking a cross above which is a bust of Jesus:
The inscription in Greek tells us what the little flask originally held: "Oil from the Wood of Life from the Holy Place of Christ" - the Wood of Life was the True Cross re-discovered by Helena, the mother of Constantine. A later ampulla found at Sant Pere de Casserres, Catalonia, shows the full crucifixion, but dates not before the 8th century (here).
The first preserved manuscript with a depiction of the crucifixion is in the Rabula Gospels, created in Syria in the 6th century - the Syriac text is signed by Rabula and dated AD 586 at the Monastery of St. John of Zagba.
What's interesting is that some scholars, such as Massimo Bernabo, now believe that the illustrations are earlier, and were taken from another Greek Gospel and inserted into the Rabula Gospels.
Egyptian monophysites tended to avoid depicting the crucifixion, but there is a Coptic magic papyrus of the 6th century with a sketch of the Crucifixion (British Library Oriental Manuscript 6796, image):
There are some other possible depictions of the crucifixion on small gems, but these are difficult to date, and seem to have been linked to the Nestorians - and so were seen as heretical by mainstream Christians. I'm going to start with the "Orpheos Bakkikos" seal which was lost during the war, but used to be in the Bode Museum in Berlin:
This shows a man crucified on a sort of anchor and is sometimes cited as 'proof' that Jesus was based on Orpheus - whether you believe Jesus was the son of God or not, he sprang from the Jewish religious milieu, and as I've pointed out repeatedly Orphism was a modern 19th century creation by scholars opposed to the power of the Vatican (here). Although conspiracy theorist like to show images of this hematite seal as proof of assorted theories, I don't know any serious scholar who does not believe it's a fake.
There are a number of magical amulets which, like the Coptic papyrus, show a crucified man, such as this bloodstone seal in the British Museum (inv MME 1986.05-01.1) allegedly found in Gaza, and possible made in the 2nd or 3rd century:
The problem with this amulet is that although the inscription says "Son, Father, Jesus Christ" it also has quite a few pagan terms on it and it seems to be a pagan exorcism object which happened to invoke anyone and everyone available, including Jesus ... rather like the Coptic Magic papyrus above, and the Jewish exorcist in Acts 19:13-17 ...
here, and there is also a spell found in the Cairo Geniza (here p. 183) ... and this was nothing new, as can be seen in Apuleius (3.17), Pliny (NH 28.6) and Lucian (Philops. 17). Jesus is invoked not as the Son of God but as a magician, something seem also in the text of a second century gold lamella which mentions assorted other pagan figures (left, Private Collection, London).
Another carnelian in the British Museum (inv MME 1895.11-13.1) is more clearly identified as Jesus by the inscription "Jesus Christ Son of God the Saviour" - it was found at Constantia in Romania, and is Christian, but probably made in Syria:
A second very similar gem from the Nott Collection is preserved in a cast at the DAI in Rome, "Jesus Christ":
Although the 12 figures flanking the crucifixion are generally described as Apostles, these were neither present at the Crucifixion nor numbered 12 after the suicide of Judas. Instead they might represent a procession of clergy, the bishops asserting their claim to temporal power because of the crucifixion and sacrifice of Jesus, which would suggest these seals were made for religious leaders. Some people would like to see these two seals as early, but they are probably 5th to 7th century (Mastrocinque)
A fourth gem seems to be later, but when it was created in the Byzantine period is far from certain, although bent knees are a late feature:
Although from Constantine onwards Christianity became the dominant religion of Rome, Christians for several centuries continued to avoid depicting 'bad' events from their history in their arts, whether it was the Crucifixion or the torture and martyrdom of their saints.
UPDATE - I got some of the images of the intaglios from a web site so wacky I didn't even want to link to it. I hadn't realised that they had in turn been scanned from Jeffrey Spier's brilliant study Late Antique and Early Christian Gems (Spatantike-Fruhes Christentum-Byzanz), or obviously I would have credited it. For those who are interested, and don't want an academic study, see also his contribution in the Kimbell's Picturing the Bible: The Earliest Christian Art (Kimbell Art Museum). I just threw this post together from images I'd been gathering for a while, but know little about, so I really recommend looking at Spier's work if you're interested in the subject as he's an excellent scholar and specialises in this area.