I am not yet a British citizen, but as an Australian resident in the country I get to vote in the forthcoming referendum on EU membership. (Unlike European citizens resident in the UK, or British citizens who have been abroad for too long.) So I take the liberty of using "we" in what follows.
The burden of proof
If we have any respect for conservatism at all, the onus has to be on advocates of leaving the European Union to make a case. Leaving the EU would be a drastic change to the UK's position in the world and there needs to be a good reason for undertaking such a leap. This is indeed the reason we are holding a referendum at all, rather than just letting parliament decide — in its gravity, this is a decision on a par with a constitutional amendment in Australia. (Where the burden is explicitly on proponents of change, and where only 8 out of 44 referendums since Federation have passed.)
Uncertainty, Tail Risks, and Optionality
The risks of continuing "as is" within the EU are fairly well understood; in contrast, there are large uncertainties in what leaving would entail. We don't even know what a UK government would attempt to negotiate — proponents of leaving have very different suggestions for this — in terms of trade, migration, finances and so forth, let alone how different partners would respond or what the outcomes would be.
As a result of this, there are significant "long tail" risks to leaving. There are some obviously terrible outcomes that are unlikely but not impossible, and the asymmetry means the greater weight of entirely unforseen risks falls to leaving.
In the event of a decision to remain, the UK would retain the option of leaving, allowing us to make a later decision when we have more information: domestic politics might make a repeat referendum in the short-term unlikely, but it would remain a possibility. In contrast, a decision to leave would be irreversible, at least in the medium-term. And any attempt to rejoin would not be under our control but would be subject to the whims of two dozen other states, so even if successful we would almost certainly end up in a worse position than we currently have.
In my opinion, these arguments carry sufficient weight that anyone uncertain of what to do should vote to remain in the EU. You may want to stop reading now.
The quality of the arguments for leaving the EU that I am seeing is just woeful.
The Leave camp is not lacking in numbers of arguments. But, as with creationism and climate science denial, the sheer number of arguments here is actually a sign of weakness. I want one or two really solid arguments, demonstrating that something of significant value will result, with high probability, from leaving; instead I'm seeing hundreds of different arguments, picked on each occasion for rhetorical effect on whoever the current listener is. Some of them are wrong, some of them are weak, some of them are inconsequential, some of them are too vague to evaluate at all, and it's actually really hard to find the ones that have some substance. (This may be just as true of arguments for Remain, but as explained above I'm not looking for an argument to remain, I'm looking for an argument to leave.)
Leaving aside the outright fabrication of numbers, a huge number of the arguments I see for leaving are based on premises that are simply false. Most commonly they conflate membership of the EU with membership of the European Single Market or the Eurozone, adherence to the Schengen Agreement, or membership of the Council of Europe and being a party to the European Convention on Human Rights. But there is also scaremongering over Turkey joining the EU — this is not only several decades away even on the most optimistic timetable, and vetoable by the UK, but getting further away every year — and over threats from Russia that would have been overblown in 1950.
A lot of the arguments are appeals to abstractions with no content: "control over our own destiny", "the right to govern ourselves", sovereignty, and so forth. These arguments are mostly so vague they work just as well as arguments for the UK leaving the United Nations or NATO, or indeed abrogating every foreign treaty it has ever entered into — or as arguments for Scotland leaving the United Kingdom, for that matter.
And then there are the "arguments" (often Daily Mail headlines) which involve matters of such triviality that they are almost laughable. Even if the EU had regulated cabbages or bus sizes in some silly fashion — and when I actually investigate these stories they turn out to be either inventions or the result of taking something completely out of context — that would be so inconsequential as to be irrelevant. (Also, it's not as if Westminster isn't capable of idiotic regulation itself, or perhaps more commonly under-regulation. And, unless you are a mindlessly doctrinaire libertarian, more general arguments about the cost of regulation on business need to take into account the benefits for individuals.)
Free Movement of Workers
I will discuss free movement of workers in a little more detail.
Having passed through the UK skilled migrant process myself, to the point of having Indefinite Leave to Remain, I can only say that it is hostile, unpleasant, expensive and time-consuming, and generally stressful. Making sure that I don't ever have to do that again will be one of the big motivations for taking up British citizenship. And the inflexible operation of the UK's immigration system sometimes involves much worse than anything I have experienced, breaking up families and destroying careers. On top of the personal and human costs, there are also economic costs: in administration and compliance but also in inefficient labour allocation.
Inflicting anything like this system on all the Europeans who want to work in the UK, and having whatever various countries require from non-EU workers inflicted on Britons working in Europe, would do real harm, both to individuals and to economies. Presumably a post-exit UK would attempt to negotiate a closer relationship with various European countries in this area than it has with Australia, in an attempt to alleviate this, but it remains entirely unclear what the UK would want, let alone what it would get. A major argument for leaving the EU is to "restore full control over immigration", which is always kept vague but surely implies imposing some kind of formal control on movement of workers from Europe — and, if getting net migration into the tens of thousands is a goal, a fairly stringently selective one. Even if other European states could be won over to (say) reciprocal agreements that only allowed people earning more than £35,000 a year to move to work, I'm not convinced people in the UK will be at all happy about a system where rich Britons get to move and work freely in Europe but poor people are excluded.
In any event, no one is going to enjoy the extra paperwork involved, except the bureaucrats who will get to negotiate two dozen new agreements and manage the resulting system. (If a post-exit UK negotiated a single agreement on movement of workers with the EU, that would certainly simplify the paperwork. However that would not only require the UK to grant Bulgarians the same status as the Dutch, something some people seem upset about, but would also cover Turks if Turkey joins the EU, which would now be possible without a UK veto...)