The short answer is "not very, but we're getting better".
In the case of particularly large Earth orbit intersecting bodies, kilometers across and bigger, they're fairly well known and tracked. We'd probably get months to years of warning. Technologically, we've probably got the capability of diverting such an object if we have enough warning, but it'd likely be the most complex human endeavor ever. Blowing it up is unlikely to help, we'd need a nice long shove from as far away as possible to divert it.
The bigger problem is smaller bodies in the tens to hundreds of metres. Some of these are known about and tracked, but plenty more aren't. The Chelyabinsk meteor was about 20 metres across and, in terms of size/density/speed/impact angle wasn't that far off being a city killer. We never saw it coming. (Ironically, a very similar but unrelated 30m object was being tracked through a near-miss that same day.)
I'd suggest that a city-killer impact with little or no warning is still quite possible. Something capable of taking out, say, a small country we'd probably have some warning of, but probably not enough. A mass-extinction event we'd probably have lots of warning for, but doing anything about it would be a very big challenge.