Yes certainly the external borders need sealing (in fact they should never have been allowed to become so porous in the first place). The ridiculous Schengen agreement has made matters far worse by allowing free movement of people - even those not entitled to be in Europe at all. Once they are in the EU they are free to go where they wish unhindered and that is why hundreds of thousands make their way from southern Italy to northern France.
However, that is not the entire answer. Many of the problems faced by the UK (pressure on housing, medical facilities, education, etc.) is a result of migration from within the EU. The principle of "free movement" was fine when the union consisted of a dozen relatively prosperous nations where wages and costs were similar. That is not now the case. Many of the later members of the EU have economies, wage and benefit levels far inferior to ours. It is natural that people from say, Poland, will want to migrate here to earn five to ten times the amount they can at home. Unfortunately this distorts the labour market in the UK. Furthermore, many of these arrivals are not net contributors to the UK's economy. Many of them have very low incomes, have to be subsidised by working tax credits, child tax credits and housing benefit and a sizeable chunk of their earnings is sent "home" with no benefit to the UK economy whatsoever. There is nothing in this for the UK - quite the reverse in fact. Add to this that by "sub-contracting" this low-skilled work out to foreigners we still have to keep those already here who will not do such work.
The Prime Minister should be honest with the electorate (and that would make a change). There is no chance whatsoever of there being any meaningful change to the free movement rules. It is one of the most important of the founding principles of the EU, is enshrined in the various Treaties and will require the agreement of all the other 27 nations for any change to be enacted. It is also the second most crowning achievement lauded by the Euromaniacs, exceeded in pride only by the single currency (which demonstrates their distorted sense of achievement).
The only way to stem the flow of people moving from mainland Europe to the UK is for the UK to leave the EU. Nothing else will do.