Republican presumptive presidential nominee Donald Trump is giving a national platform to parents of victims of crimes by illegal immigrants, inviting them to rallies and telling their tragic stories to boost support for hard-line immigration policies.

Mr. Trump is drawing intense media attention to what he says is a public-safety issue caused by illegal immigration. But the families also help put a sympathetic face on Mr. Trump’s attacks on some Mexican immigrants as criminals and his plans to build a wall along the southern...