There seem to be two strands of opinion which encourages people on the left to vote Leave. The first is an intense dislike of what has happened in the the Eurozone: see Larry Elliott for example. The second is a wish to take seriously working class concerns over immigration.
I share the intense dislike over what has happened in the Eurozone (EZ). You only need to read, for example, what I have written about Greece to see that. Yet I cannot see how UK exit from the EU will change for the good how the EZ works. Crucially why should it change the current politics of Germany that has been so important in many of the bad decisions the EZ has made. As we are not part of the EZ, it is difficult to see how UK exit will help its demise, if that is what you want.
As we will be leaving the EU and not the EZ, the message Brexit sends to Europe is not that the EZ is fundamentally flawed but rather that the UK is incapable of being part of any cooperative agreement among European countries. I am internationalist by nature so I do not want that. The idea that Brexit will shock the EZ into mending its ways, then thank us for showing them the light and invite us to rejoin whatever is left is pure fantasy.
There is one way the EZ crisis has impacted on the UK, and that is migration. It seems reasonable to assume that one reason immigration into the UK from the EU is currently high is because of considerable youth unemployment in many EZ countries. Which brings us to immigration concerns.
Take this from Kate Hoey for example. She seems to be arguing that free movement in the EU is a means of keeping down the wages of the low paid. The first point to make is if labour mobility keeps wages down in the destination country, it should increase wages and/or reduce unemployment in the country the migrant came from. As migrants move from lower to higher wage countries, then migration tends to equalise incomes. This should normally count as a plus from a left wing perspective.
But does migration have this effect on the receiving country? There are many reasons why it might not, and the evidence does not point to strong negative effects on wages. But I think the case for migration is stronger than that. A pretty robust finding is that migration at the kind of levels we are now seeing does not do any harm to GDP per head, and could improve it. Another robust finding is that current migration benefits the public finances. This is both important and pretty obvious..
Migrants tend to be young, healthy and working. They provide more in terms of resources than they take out by using public services. I remember having a conversation about this with someone who lived in Spain. He said if anyone should be angry about free movement it is Spain, in having to take lots non-productive British pensioners who will be a burden on Spain’s health service. This was also one reason why Merkel could be so open to refugees: Germany is really worried about the implications of their aging population. 
If migration falls following Brexit (a big if), and if we add in the other negative effects of Brexit, we will have a large increase in the government’s budget deficit even at full employment. Given government policy on how holes in the deficit are to be filled, this NIESR analysis suggests you are talking about large hits to the lower paid.
None of this is to deny real grievances about reduced public services and lower pay. Once again I do not think I and many others could be accused of keeping quiet about the stupidity of current austerity. But just as the grievances are real, it is also important to understand the real causes.
A consistent result in polls is that migration is a concern mainly in areas where there is little of it. In this poll only 19% of people say they believe immigration has had a negative effect on them personally. (More believe they have experienced positive effects.) Yet over 50% say it has had a negative impact on the NHS. In reality the NHS is in crisis because of government policy, not because of immigration. But to know that you need to talk to the experts and look at the figures. Instead most people will just have read the newspapers, many of which have taken every opportunity to play up the ‘threat’ of migration. As Paul Mason nicely puts it, you can tell this is a fake working class revolt from looking at those leading it.
This is part of a more general point on migration. Migration appears to be at worst neutral and probably beneficial for medium term UK prosperity, so if there are problems for some groups that can be fixed. At which point those on the left say that is naive because it will not happen. That is a reasonable point, but the same point applies in spades to the immediate aftermath of Brexit, which will amount to yet another shift to the right. The regulations that those running the Leave campaign want to ‘free’ us from are those protecting workers and the environment.
To make this clear, think of two realities (economists would call them states of the world). One (the status quo) involves those currently in charge remaining in charge. The second (left renaissance) involves left wing governments coming to power in the UK and elsewhere. It seems to me that if you stay in either one of these two realities, Brexit is clearly bad. (For a similar argument, see this by Alex Andreou.)
Under the status quo, the EU is currently a moderating influence on this right wing UK government. A post-Brexit Boris Johnson government would take away worker rights that the EU currently ensures and, as the deficit deteriorates, cut welfare benefits including tax credits. This alone will dwarf any benefit working people might get from less immigration. Under a left renaissance migration benefits the economy as a whole as well as those who migrate, and these gains could be fairly distributed. My feeling is that some Brexit supporters on the left move freely between these two realities in order to justify a vote for Brexit.
 One fair point made in comments on earlier posts is that migrants need to be housed, so resources are used to provide additional housing. (More generally new labour needs new capital of various kinds.) But housing also represents why migration can be beneficial to everyone. We need to build more houses anyway, but according to the industry we have an acute shortage of skilled construction workers. Of course this is an indictment of UK training programmes, but that cannot be turned around quickly. For those with construction skills, it is surely better to be building houses where they are really needed than being unemployed in places they are not.