The problem with putting all "immigration" in a single bucket is that it leads to all sorts of confusion about who might be affected by what incentives. Non-immigrants also tend to be completely unaware of how the system works. Each EU country has its own rules for non-EU immigrants; I'll use the UK points-based system as a guide. (Yes, the system is already points-based, so anyone promising to "introduce a points based system" is misleading you.)
We can also see the statistics for how many people use each kind of visa.
Tier 1: "entrepreneur" visas. Rich people only. Unlikely to be affected by minimum wage.
Tier 2: general work permits. Mostly require a specific job with an income of £30k (not affected by minimum wage), apart from some specialties which require £20k. Minimum wage is about £15k a year. The NHS is heavily dependent on recruiting nurses through this system, for example.
Tier 3 was never implemented.
Tier 4 is students. No relevance of minimum wage.
Tier 5 is temporary and youth workers. The minimum wage might be relevant here.
Further categories of non-visa entrants are family reunification (I don't think the minimum wage affects who people marry, and in any case there's a minimum income requirement for a spouse visa), and asylum. Asylum seekers are not allowed to work while their claim is being processed, may be subject to detention, and have an uphill burden of proof.
Almost none of these are "cheap" labour, except some of the tier 5 fruit pickers and other seasonal workers.