There is at least one study (PDF) that did not find a statistical difference. In their own words:
“We find no statistically discernible difference in violent crime rate, rape, or property crime across the cities,” the researchers concluded. “Our findings provide evidence that sanctuary policies have no effect on crime rates, despite narratives to the contrary.”
One of the study’s co-authors, Ben Gonzalez O’Brien, a professor of political science at Highline College, told us via email that this study and others have found no support for Trump’s claim.
“In past statements, Trump has cited individual instances of crime, such as the Kathryn Steinle shooting in San Francisco, rather than any evidence that sanctuary cities ‘breed crime,'” Gonzalez O’Brien said. “If this was the case we would expect to see higher crime rates in sanctuary cities when compared to cities with similar demographic characteristics, or an increase in crime after a sanctuary declaration was made. In our research, we have found no support for either of these propositions. There is no generalizable or statistical evidence that crime increases after a city becomes a sanctuary.”
Of course, they are talking about something different than what Donald Trump is claiming. They are talking about broad overviews in non-experimental context (i.e. instead of dividing cities randomly into sanctuary/non-sanctuary, the cities divide themselves). Trump is discussing individual events, e.g. Kate Steinle's death. If that particular illegal immigrant who was not deported as a result of a sanctuary policy had been deported instead, that particular victim would likely still be alive.
According to the first paragraph of that quote, they've also debunked the competing narrative that sanctuary cities have less crime because immigrants are more willing to work with authorities. Or the narrative that sanctuary cities have more reported crime (if not necessarily more actual crime) because immigrants are more willing to work with authorities. The net impact on crime is not statistically significant in either direction.
Perhaps multiple narratives are true but they cancel each other. Or perhaps this study is flawed (in either direction). Crime studies are inherently difficult, as laws don't lend themselves to experimentation.