Right, well, that happened.
To say the Naz Shah incident was a simple decision that was appallingly badly handled would be an immense understatement. As for Livingstone... Christ!
Leaving all that to the side there's the question of whether Labour have an antisemitism problem. This is a sensitive and thorny issue and one to which I don't have a concrete answer. But, to give some clarity to what is being discussed, there's two ways of how it's defined as a problem.
The first is to define it in terms of numbers and ask 'Does Labour have more antisemitic members than other parties on average?' This one is may well be true, but it's likely to be true for reasons that are independent of Labour. Put simply, Labour is the most left mainstream party; left parties tend to be less pro-Israel  than other parties; so antisemitism will be more likely to join Labour than they will another party. In this sense they might have a problem, but it would be the same problem that the Conservatives have in attracting more anti-Muslim people, or general racists (though this might be siphoned a bit by UKIP's rise), to the party.
In this instance there isn't an awful lot that can be done. Background checks aren't really feasible for everyone who joins the party (though you'd like to think they'd be better at doing it for people who are standing to be MPs...) Being vigilant and suspending and expelling when it comes to light is about all that can be done.
The second way of defining is to ask whether Labour has a systematic problem, which is to say 'Do Labour's actions cause antisemites to join the party?' This one is trickier. I suspect it's not correct; despite the way matters have been handled the leadership has been clear about their views on this and their bungling of this matter has been no different to their bungling of other matters. Evidently they need to get much better and more decisive at responding to antisemtism when it comes up, but that is true of the handling of much of what they're doing. There might a be a problem in that, in the past, Labour has been a little too eager to do dog-whistles on issues (immigration, foreign workers) to court votes and this stuff can very obviously bleed into other aspects ('hey if it's okay to be down on x it must be okay to be down on y too!'). This is a slightly separate issue, as its a general one to British politics at the moment. The point being that once you start to lift the lid on Pandora's box there's all sorts of unintended consequences that might start flying out.
As a last word, I'll just make a general point that's not connected to the above. On the question of Israel and antisemitism there's an analogy that can be drawn with immigration and racism. There are some legitimate criticisms that can be made about immigration policy; this doesn't make the people doing this racists. However it is also true that lots of racists will criticize immigration because they are racists and are using concern about immigration as a cover for their racism. The same is true about Israel and antisemitism. There are legitimate criticisms that can be made about Israel's policies in the West Bank and Gaza; doing so does not make people antisemitism. However it is also true that antisemitism will criticize Israel because they are antisemitism and are using concern of Israel's policies as a cover for that.
Distinguishing between the two is often quite hard, but that's why there's a need for this to be dealt with seriously and maturely. Conflating the two, in the easy manner that both left and right are often too happy to indulge in, does not help the issue at all.
 I'm not going to launch into a discussion about the differences between Zionism and Judaism - it's a largely political thing historically and is not relevant to this discussion. For what it's worth the basic run-down of it is: Zionists were Jews who were fed-up of being persecuted and waiting for God to lift his finger and do something, so they decided to take matters into their own hands and found a state for themselves. To create this sense of 'nationality', for want of a better word, they went back and associated themselves with the stories of Hebrew warriors (as a more assertive group, to the perceived passive acceptance of the Jews of their persecution by Christians and others) and were quite explicit about distinguishing themselves from Jews, calling themselves Hebrews. Naturally the distinction isn't quite so clear now as, after that little thing called the Holocaust, most Jews quite naturally felt that they'd had enough and wanted a place of their own.