The theological problem is to explain why Jesus apparently trusted Judas, when he ought to have known better. There are a lot of possible answers! Some have tried to find symbolic or exemplary meaning in his actions - a lesson for the future Church. Others have centered the discussion around the character of Jesus, perhaps reaching similar conclusions for different reasons: framing Jesus principally in terms of love rather than as a ritual performer.
Augustine says in his Tractates on John 50.10-11 that Jesus trusted Judas in order to show the Church should be patient and tolerant even of thieves:
What lesson then, my brethren, did our Lord Jesus Christ wish to impress on His Church, when it pleased Him to have one castaway among the twelve, but this, that we should bear with the wicked, and refrain from dividing the body of Christ? [...] Why gave He admission to a thief, save to teach His Church patiently to bear with thieves? But he who had formed the habit of abstracting money from the bag, did not hesitate for money received to sell the Lord Himself. But let us see what answer our Lord gave to such words. See, brethren: He does not say to him, You speak so on account of your thievishness. He knew him to be a thief, yet did not betray him, but rather endured him, and showed us an example of patience in tolerating the wicked in the Church.
To this, Thomas Aquinas adds two further suggestions (Lectura super Evangelium Sancti Ioannis 12.1). The first is that Judas was given access to the common fund so that he could satiate his greed there, rather than condemn himself further by robbing others; Aquinas finds this doubtful, since the greedy are never truly satisfied. This matches his Aristotelian ethic of virtue and habit - indulging in theft only reinforces the "bad habit" of having a tendency to steal. He would certainly agree with you that it would be irresponsible for Jesus to not only put Judas where he could be tempted, but even to encourage him in his sinful habits. The second suggestion from Aquinas, which he prefers, is that Jesus sought to teach the Church that spiritual things are more important than temporal things: by giving the money over to Judas, he was showing how little it mattered.
Origen (Commentary on John 32) is of the same opinion as Augustine as far as the lesson - he argues that Jesus showed trust as a deliberate act of love. Moreover, he says that Judas should not be seen as simply an evil person, but as someone who had both good and bad in him (he was a disciple, after all), was continually presented with moral choices, and on several important occasions made terrible decisions. In Origen's view, Judas had the genuine free will to steal and betray, or not, and so Jesus was justified in trusting him. From 32.161-162 (trans. Ronald Heine; Catholic University of America Press, 1993):
I would not think that he would have been entrusted with the money-box if he were a thief from the beginning. He was trusted with it, therefore, because he was worthy of being trusted, although it was foreknown that he would fall away. And he was so great a man of Christ's peace that Jesus once had high hopes in him, as a good apostle, for hear the words, "In whom I hoped" [from Psalm 41:9, which is quoted in John 13:18].
If Judas' evil had been obvious to Jesus' disciples it would have been known who was to betray the teacher, since Jesus had said, "One of you will betray me." But now the disciples look "at one another, doubting of whom he spoke" [John 13:22]. Perhaps indeed the apostles were ashamed to suspect anything wicked of Judas because of his previous worthy deeds. It may have been, too, that Judas did not belong totally to evil, even though the devil had already put it in his heart that Judas Iscariot, son of Simon, should betray him [John 13:2]. It was because there was still a remnant of good choice in him that, when he saw that Jesus was condemned when "they bound him and led him away and delivered him to Pilate the governor" [Matthew 27:2] he repented and returned the thirty pieces of silver to the chief priests and elders, saying, "I have sinned in betraying innocent blood" [Matthew 27:3-4].
Origen's exegesis does not see Judas as a moustache-twirling villain, or as a kleptomaniac who couldn't help himself from stealing, but as a very human figure. The fact that he returned the silver shows that he had the capacity to do good. Since the disciples were unaware of Judas stealing from the common purse, his acts must have been more like incidental pilfering than outright theft of the whole amount. Perhaps, along these lines, we can say that Jesus's trust in Judas was not too financially disastrous.
The absolute opposite view on free will is held by John Calvin, who says (Commentarium in evangelium Ioannis) that firstly, the ways of God are mysterious (an admission that he has no real answer), but secondly, Judas was predestined to be lost: Jesus placed an untrustworthy person among the disciples because the plan required one of them to betray him. Calvin's Judas is "required" to be evil and then to be justly condemned, so Jesus can perfectly well put Judas in a position of temptation, knowing that he will fail. This resonates with the opinion of John Chrysostom (Homilies on the Gospel of John 65), who says that Judas was given charge of the purse so that when he betrayed Jesus for thirty pieces of silver, he wouldn't have the excuse that he really needed the money.
Perhaps characteristically, Karl Barth (Church Dogmatics, 35.4, "The Determination of the Rejected") welds all of these ideas, and more, into a single incredibly dense lump of thought. He sees the sin of Judas, in betraying Jesus, principally as an exploitation or rejection of the sacred. This is mirrored in his eventual suicide, in his attitude to the anointing in this passage, and in his love of money. But he is still a human being and a disciple - and all of the disciples have their bad moments, as they fail to live up to the expectations of Jesus. So there is a sense in which Judas provides an example for us, not only personally but in the life of the Church (his replacement by Matthias is critical for this - a purging of communal failure). But the more important statement, for Barth, is that Jesus is on Judas's side, even if Judas is not on his side. We are meant to imitate Christ in love of God and one another, even our "enemies", just as he put trust in the one who would betray him.