Romans 16:19 (NKJV)
Sensitivity can be a very subjective thing. Some people are more sensitive to light than others while some people are more sensitive to certain smells. In both cases, the person with the heightened sensitivity is more aware of, and more influenced by, the things they are sensitive to than other people are to those same things. Some sensitivities can be developed, like sensitivity to certain sounds or tastes, because of someone's repeated exposure to and experience with these things. While many types of sensitivity seem to occur naturally in a person, some sensitivities can be developed.
The things we spend our time with will develop a sensitivity, an awareness, within us. If, for example, someone was to listen to classical music all day long it would become easier and easier for them to recognize certain selections of music and their composers. This same kind of thing can, and should, happen with Christians. As we spend more time in God's Word, talking to God in prayer, and being with other like-minded believers our sensitivity to the things of God can increase. Spending less time with these things would make us less sensitive to godly things.
As with most things, we can develop in our sensitivity to what is good and God honoring. We can develop an understanding of how God sees things and what He considers to be right and wrong, rather than having our concepts of these things being influenced by society and other people. By our continuing to spend time in God's Word and in prayer we can develop a sensitivity and awareness to be wise towards what is good, as Romans 16:19 says. This helps us to follow God and live a life that is not only satisfying, but one that is pleasing to Him as well.