A number of surveys have asked if various political figures "care about people like you." Here are the figures for presidential candidates. When possible, they are taken from surveys shortly before the election:
Cares Doesn't
Trump 46% 54% Nov 2016 (post-election)
H. Clinton 47% 51% Aug 2016
Trump 37% 60% Aug 2016
H. Clinton 46% 51% May 2016
Trump 42% 55% May 2016
H. Clinton 47% 52% May 2015
Obama 61% 35% Jan 2012
Obama 57% 41% Oct 2011
Obama 66% 33% Oct-Nov 2008
McCain 54% 44% Oct-Nov 2008
GW Bush 50% 46% Oct 2004
Kerry 54% 40% Oct 2004
Dick Cheney 40% 50% July 2004
GW Bush 51% 41% Oct-Nov 2000
Gore 58% 34% Oct-Nov 2000
GHW Bush 51% 39% May 1991
Dukakis 55% 33% Nov 1988
GHW Bush 42% 48% Nov 1988
GHW Bush 52% 37% Sept 1988
Reagan 56% 37% Oct 1984
Mondale 70% 22% Oct 1984
Carter 55% 32% Nov 1979
Ford 47% 35% July 1976
Carter 48% 22% July 1976
Trump did not do well in terms of being seen as caring about "people like you"--in fact, he did worse that anyone else on the list, despite tough competition from Dick Cheney (it's not comprehensive, so I can't say it was the worst ever). So what we can learn from Trump is that it's possible to win even if most voters think you don't care about people like them.
There are a couple of interesting patterns, First, Democrats consistently are more likely to be seen as caring about "people like you." Someone (I think Butler and Stokes, Political Change in Britain) observed that in Britain, Labour was seen as more concerned about ordinary people and the Conservatives as more effective. Second, Hillary Clinton did poorly for a Democrat. This seems to be specific to her rather than a downward trend for Democratic candidates.