Q:

Does killing civilians increase the difficulty of the police response?

Occasionally when I accidentally kill civilians be it collateral damage or I just don't want to deal with them... Bain sometimes says: 

Hey! They're gonna bring in the big guns if you kill civilians!

Or 

They're gonna come down harder on us if you kill civilians!

Does killing civilians increase the difficulty / produce a tougher police response? 

A:

From when I used to play I only remember them being useful as hostages and nothing more. Looking back on the wiki it seems I am mostly right.
To directly answer your question: 

"Does killing civilians increase the difficulty / produce a tougher
  police response?"

The answer is no. The actual game difficulty does not increase, and no "big guns" show up with the cops.
Killing civilians is one way to prevent them from running around setting off your traps, or alerting the police earlier, however there are consequences:
"The players are discouraged from killing civilians; collateral damage will immediately deduct funds from the player's spending cash (regardless of whether the heist is completed) and add a trade penalty to that player if he or she goes into custody."
So in short, the big penalty for killing the innocent people is a reduction in the money you earn, and making it harder to trade hostages if a teammate or yourself is taken into custody.

