The phrase "delicate sensibilities" refers to someone being easily offended or upset by things that others may not find offensive or upsetting. Full definition
For one with more delicate sensibilities, I would point them to the subprime mortgage market in 2006, of which I wrote a timely piece. (alephblog.com)
Your feet may end up leading you to the Church of St Francis, which plays host to the grave of Portuguese mariner Marco de Vasco who first ventured to this part of the world in 1498, as well as evidence of the more delicate sensibilities of other travelling Europeans. (greatrail.com)
With the exception of the fact that somebody has been bumped off, the cozy tends to be light in tone; that is, it is crafted so as not to offend delicate sensibilities. (thebalancecareers.com)