I fear that many - perhaps even most - non-believers would answer that it doesn't. Many would even suggest it does the opposite - that Christians are hypocritical, judgemental, self-righteous, out of touch, etc. Mahatma Ghandi for example, who had many sympathies with the Christian faith but remained a committed Hindu all his life, is alleged1 to have once said, "I like your Christ, I do not like your Christians".
For myself I've known some Christians who seem to be genuinely "saintly", in the best popular sense of that term - kind, caring, generous, patient, loving, big-hearted people. I've also come across some very saintly people who are not Christians, but my general experience of most people is that they are a mixed bag of tricks. I can sometimes be shocked by a person's arrogance or selfishness or their apparently shallow or short-sighted attitude, and then shortly afterwards be amazed at the same person's kindness or sensitivity in a different situation.
What about me though? Does being a Christian make me a better person? This is a hard question to answer. For one thing, I was raised in a Christian family and made my first commitment to Jesus when I was four and a half. It's difficult to remember what I was like before this and given the age difference between now and then I hardly think it would be a fair comparison in any case! I was brought up with certain moral standards. For example I was taught to respect and obey authority (this may have been over-emphasised...), always to tell the truth, not to swear, not to engage in physical violence, not to steal, etc.
It wasn't until my early teens however, when I began to question my faith and had my first major experience of God that I can remember, that I first began to realise how selfish I was. God showed me all sorts of unpleasant things about my heart that didn't seem to have been touched by all those rules I'd learned to follow. God also showed me - at the same time - how much He loved and accepted me, which felt great at the time, but which I began to lose sight of as I realised that all of those bad attitudes didn't seem to be changing very much!
For a long time I've compared myself and other Christians to those who don't seem to know God or Jesus and wondered if it really does make any difference. I can always find ways to compare myself unfavourably with others. Likewise I can usually find some way in which I think I excel - but even as I am internally praising my moral superiority over some poor, unenlightened heathen, I notice my own judgemental self-righteousness and am knocked back down a peg again!
What can I say now? That I'm still very much in process and have learnt that following Jesus is a journey. That without love and acceptance I'm unable to change - I just get discouraged and frustrated, which ends up encouraging and re-inforcing the same negative attitudes and behaviours that I'm trying to replace! That I suffer from the same moral weaknesses and afflictions as everyone else but have learnt that discipline and good habits can make a difference. Finally, that I can't do it alone - I need help from God and from others who are similarly committed to developing this way of life.
Does being a Christian make me a better person? I hope that it slowly is doing - but only time, and those around me, can really say.
1 This has been disputed - see here - but I still think it unfortunately sums up many people's feelings towards the Christian faith.