There seems to be a climate of fear in the US, but that isn't necessarily due to President Obama. Since 9/11, the US government has been either pandering to a scared population that believes themselves in imminent threat of death or have exploited the same fear in order to increase federal power.
Yes there is a large climate of fear under Obama in the US. Anyone who does not like what he stands for and what he is doing is causing this. There would likely be some fear under any President but Obama is causing more with his actions as the President.